aa r X i v : . [ m a t h . DG ] M a y Université d’Abomey-Calavi (UAC), Bénin
The Abdus-Salam International Centre for Theoretical Physics (ICTP), Italy
Institut de Mathématiques et de Sciences Physiques (IMSP), Bénin
Order n : 43/2010 PhD ThesisOption : Differential GeometryTitle : On the Geometry of Cotangent Bundles of Lie Groups
ByBakary Manga Defended in June th , Supported by The Abdus Salam International Centre for Theoretical Physics (ICTP), Italy edication
To My Sister Binta Manga
On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 cknowledgements
As prelude to their works, authors often display a warning, a kind of umbrella, fortunatelyopened to protect themselves from possible backslash of their statement. As an author, ifI was allowed to do the same, I will state the following: I acknowledge that It will be quitehard to find convenient words which can express my thanks to all those who helped me toachieve the dream of writing a PhD thesis. For those whose names are not written here,please be indulgent with me and accept my apologies.I wish long life to the
Institut de Mathematiques et de Sciences Physiques (IMSP) , theinstitute in which this thesis has been achieved.The relations between
Professor Jean-Pierre Ezin , director of the IMSP (1988-2008)and me have never been always easy. Nevertheless, Prof. Ezin has contributed to makeme become a better person by allowing my personality, even overflowing at times, to beexpressed. I am grateful to you for that and as well as for your strong and paramountbattle related to support fundamental sciences in Africa. Thank you commissary . Professor Joël Tossa , has been for me more than an advisor. With me, he has beenalways available and patient. He constantly did his best in order to provide me with so-lutions whenever my working and living conditions were tough. Our relations have beenintensified over years such that it is becoming difficult for me not to have it around me.For sure, he will be one of those persons that I will miss a lot once I am back in Senegal.I owe him quite a lot.I was very lucky to have
Professor Alberto Medina of the University of Montpellier II(UM II) as another advisor of my thesis. In 2005,
Professor Medina invited me for a stayin his research group whose niche is the Homogeneous spaces (Lab. of Geometry, Topologyand Algebra (GTA), Department of Mathematics, UM II). He has always been there forme and has boosted me every time the spectrum of doubt and abandon was in my mind.At times, it is hard to ask to our thesis’s advisers some type of questions. Such situationshappen mainly because in our thoughts, we tend to believe that those questions are nottough enough and moreover, we do not want to show them our weaknesses. Hence, there isa need to have someone to whom our doubts and stupidities can be conveyed without anyhesitation. In Doctor André Diatta, I found that person. I was more that lucky to have Mr. Ezin is nowadays commissary in charge of Human Resources, Sciences and New Technologies ofthe African Union.
On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ii him and never, he complained. Most of the results available in this thesis would have notbeen possible without his energetic support. Several of the results within this thesis areobtained in collaboration with him and some of them will be published in joint papers. Bigbrother, I am your servant! Hermione Gandjidon and
Rekiath Yasso have had to learn the L A TEX in order to becomemy "Secretaries". I thank both of you.By being at the IMSP, I met a huge number of researchers, Physicists and Mathemati-cians. All interesting interactions and exchanges that we had had helped me to become aMathematician. I thank you all.At the campus of the "Ecole Normale Superieure Nadjo" where live PhDs and Engi-neering students and where people call me
The Old , I met guys funnier than me. I will missthe afternoon soccer’s games of Tuesdays and Fridays through which I had opportunitiesto demonstrate to young people that
The Old still has some youth in him.I spent many years in this wonderful country and I would like to emphasize that thehospitality of the Beninese people has nothing to envy to the famous Senegalese "Teranga" ("Teranga"= hospitality). Here is an opportunity to express my gratitude to all the peopleof Benin.To the Ly ’s family who adopted me and the Senegalese community of Porto-Novo, Iwould like to express my sincere thanks.I would also like to convey my profound gratitude to families Vigan, Dossa and Hounke-nou all from Porto-Novo.I am grateful to Professor Augustin Banyaga who has accepted to be a referee for mythesis. Professor Léonard Todjihounde also provided a report for this thesis. Many thanksto him.All my studies at the IMSP have been funded by the
International Centre for TheoreticalPhysics (ICTP) through the
ICAC-3 program . My trip and stay to Montpellier was fundedby
SARIMA project . Thanks a lot for your financial supports.At last but not at least. My family has endured my absence for years. They have showna lot of patience and devotion. Thank you so much dear parents, brothers and sisters. Myfriends from Senegal and elsewhere have been of a tremendous and immeasurable support.At several occasions, they have represented me efficiently where I was not able to be. Theyencouraged and motivated me all along the way. Thanks to
Aïssatou Massaly, MohamedBadji, Aliou Sarr, Anita Hounkenou, Coumba Tiréra, Hervé G. Enjieu Kadji, YoussoufMassaly, Samba Mbaye . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 bstract
In this thesis we study the geometry of cotangent bundles of Lie groups as Drinfel’d doubleLie groups. Lie groups of automorphisms of cotangent bundles of Lie groups are completelycharacterized and interesting results are obtained. We give prominence to the fact that theLie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Liegroups (Theorem 2.3.2). In the cases of orthogonal Lie algebras, semi-simple Lie algebrasand compact Lie algebras we recover by simple methods interesting co-homological knownresults (Section 2.3.6).Another theme in this thesis is the study of prederivations of cotangent bundles of Liegroups. The Lie algebra of prederivations encompasses the one of derivations as a subalge-bra. We find out that Lie algebras of cotangent Lie groups (which are not semi-simple) ofsemi-simple Lie groups have the property that all their prederivations are derivations. Thisresult is an extension of a well known result due to Müller ([64]). The structure of the Liealgebra of prederivations of Lie algebras of cotangent bundles of Lie groups is explored andwe have shown that the Lie algebra of prederivations of Lie algebras of cotangent bundleof Lie groups are reductive Lie algebras.Prederivations are useful tools for classifying objects like pseudo-Riemannian metrics([9], [64]). We have studied bi-invariant metrics on cotangent bundles of Lie groups andtheir isometries. The Lie algebra of the Lie group of isometries of a bi-invariant metric ona Lie group is composed with prederivations of the Lie algebra which are skew-symmetricwith respect to the induced orthogonal structure on the Lie algebra. We have shown thatthe Lie group of isometries of any bi-invariant metric on the cotangent bundle of any semi-simple Lie groups is generated by the exponentials of inner derivations of the Lie cotangentalgebra.Last, we have dealt with an introduction to the geometry the Lie group of affine mo-tions of the real line R , which is a Kählerian Lie group (see [53]). We describe, throughexplicit expressions, the symplectic structure, the complex structure, geodesics. Since thesymplectic structure corresponds to a solution of the Classical Yang-Baxter equation r (see[28]), we also study the double Lie group associated to r . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ésumé en Français
Cette thèse est une contribution à l’étude de la géométrie des fibrés cotangents des groupesde Lie et des espaces homogènes.Nous caractérisons complétement les groupes des automorphismes des fibrés cotangentsdes groupes de Lie et montrons qu’ils sont des groupes de Lie super-symétriques (Theorem2.3.2). Dans le cas particulier d’un groupe de Lie orthogonal, c’est-à-dire un groupe deLie muni d’une métrique bi-invariante, nous utilisons la métrique pour réinterpréter lesrelations et retrouver des résultats connus de cohomologie.Le fibré cotangent T ∗ G d’un groupe de Lie G peut être identifier au produit cartesien G × G ∗ , où G ∗ est l’espace dual de l’algèbre de Lie G de G . On peut alors munir T ∗ G de lastructure de groupe de Lie obtenue par produit semi-direct de G et G ∗ via la représentationcoadjointe. Cette structure de groupe de Lie fait de T ∗ G un double de Drinfel’d.Nous avons également étudié les préderivations des algèbres de Lie des fibrés cotangentsdes groupes de Lie. Nous montrons que l’algèbre des préderivations des algèbres de Lie desgroupes de Lie fibrés cotangents des groupes de Lie sont réductives. Müller a montré quetoutes les préderivations d’une algèbre de Lie semi-simple sont des dérivations (interieures).Nous étendons ce résultat en montrant que si un groupe de Lie est semi-simple alors toutesles préderivations de l’algèbre de Lie de son fibré cotangent sont des dérivations quoi quele fibré cotangent soit non semi-simple (Theorem 3.4.1).Un autre thème abordé dans cette thèse est l’étude des métriques bi-invariantes sur lesfibrés cotangents des groupes de Lie. Nous caractérisons toutes les métriques bi-invariantessur les fibrés cotangents des groupes de Lie et étudions le groupe de leurs isométries.L’algèbre de Lie de ce groupe d’isométries n’est rien d’autre que l’algèbre de toutes lespréderivations de l’algèbre du fibré cotangent qui sont antisymétriques par rapport à lastructure orthogonale induite sur l’algèbre de Lie du fibré cotangent.Enfin, nous avons fait une introduction à la géométrie du groupe de Lie des transfor-mations affines de la droite réelle. Nous donnons des expressions explicites d’une formesymplectique, d’une structure affine, des géodésiques de cette structure affine. La formesymplectique donnant lieu à une solution des équations Classiques de Yang-Baxter, nousavons également étudié le groupe de Lie double de Drinfel’d du groupe des transformationsaffines de la droite réelle. On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ontents
Dedication iAcknowledgements iiAbstract ivRésumé en Français vGeneral Introduction 11 Invariant Structures on Lie Groups 4 D := T ∗ G . . . . . . . . . . . . . . . . . . . . 232.3.1 Derivations of D := T ∗ G . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 A Structure Theorem for the Group of Automorphisms of D . . . . . 26 On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010
ONTENTS vii ξ and Bi-invariant Tensors of Type (1,1) . . . . . . . . . . . . 30Adjoint-invariant Endomorphisms . . . . . . . . . . . . . . . . . . . 30Maps ξ : G ∗ → G ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.4 Equivariant Maps ψ : G ∗ → G . . . . . . . . . . . . . . . . . . . . . 322.3.5 Cocycles G → G ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.6 Cohomology Space H ( D , D ) . . . . . . . . . . . . . . . . . . . . . 352.4 Case of Orthogonal Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 362.4.1 Case of Semi-simple Lie Algebras . . . . . . . . . . . . . . . . . . . 392.4.2 Case of Compact Lie Algebras . . . . . . . . . . . . . . . . . . . . . 412.5 Some Possible Applications and Open Problems . . . . . . . . . . . . . . . 432.5.1 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5.2 Invariant Riemannian or Pseudo-Riemannian Metrics . . . . . . . . 462.5.3 Poisson-Lie Structures, Double Lie Algebras, Applications . . . . . 472.5.4 Affine and Complex Structures on T ∗ G . . . . . . . . . . . . . . . 47 T ∗ G . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.1 Prederivations of T ∗ G . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.2 A Structure theorem for the Lie group Paut ( T ∗ G ) . . . . . . . . . . 533.3.3 Maps ξ : G ∗ → G ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Orthogonal Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.1 Maps α , β , ψ , ξ in orthogonal Lie algebras . . . . . . . . . . . . . 613.4.2 Semi-simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 643.4.3 Compact Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5 Possible Applications and Examples . . . . . . . . . . . . . . . . . . . . . . 703.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.2 Possible Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 73 T ∗ G . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.1 Bi-invariant Tensors on T ∗ G . . . . . . . . . . . . . . . . . . . . . . 804.3.2 Orthogonal Structures on T ∗ G . . . . . . . . . . . . . . . . . . . . . 834.3.3 Skew-symmetric Prederivations on T ∗ G . . . . . . . . . . . . . . . . 844.4 Case of Orthogonal Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 854.4.1 Bi-invariant Metrics On Cotangent Bundles of Orthogonal Lie groups 854.4.2 Skew-symmetric Prederivations . . . . . . . . . . . . . . . . . . . . 87 On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010
ONTENTS viii R . . . . . . . . . . . . . . . 944.6.2 The Special Linear Group SL (2 , R ) . . . . . . . . . . . . . . . . . . 984.6.3 The -dimensional Oscillator Lie Group . . . . . . . . . . . . . . . . 100 R R and its Lie algebra . . . . . . . . . . . . 1045.2.2 Symplectic and Affine Structures on the Affine Lie group . . . . . . 1045.2.3 Geodesics of ( G , ∇ ) at the unit . . . . . . . . . . . . . . . . . . . . 1065.2.4 Integral curves of left invariant vector fields on G . . . . . . . . . . 1105.3 Double Lie groups of the affine Lie group of R . . . . . . . . . . . . . . . . 1125.3.1 Double Lie group of the affine Lie group of R . . . . . . . . . . . . 1125.3.2 Connection on the double of the affine Lie group . . . . . . . . . . 1155.3.3 Geodesics of ( D ( G, r ) , ∇ ) . . . . . . . . . . . . . . . . . . . . . . . 1175.3.4 Integral curves of left-invariant vector fields on the double Lie groupof the affine Lie group of R . . . . . . . . . . . . . . . . . . . . . . . 1185.3.5 A Left Invariant Complex Structure On The Double of The AffineLie group of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 General Conclusion 123References 125
On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 eneral Introduction
This thesis is a contribution to the large task of studying the geometry of cotangent bundlesof Lie groups and the geometry of homogeneous spaces.One of the greatest importance of the cotangent bundle T ∗ G of a Lie group G is thatit is a symplectic manifold on which G acts by symplectomorphisms with a Lagrangianorbit. A symplectic Lie group is a pair ( G, ω ) consisting of a Lie group G and a closednon-degenerate -form ω which is invariant under left translations of G . If identified withthe Cartesian product G × G ∗ , the cotangent bundle T ∗ G possesses a Lie group structureobtained by semi-direct product T ∗ G := G ⋉ G ∗ using the coadjoint action of G on the dualspace G ∗ of the Lie algebra G of G . This Lie group structure together with the Liouvilleform is not a symplectic Lie group. But, if G carries a left-invariant affine structure, thenits cotangent bundle carries a symplectic Lie group structure. This particular Lie groupstructure, sometimes called "cotangent" , is obtained by taking the semi-direct product of G and the dual space G ∗ of its Lie algebra G by means of a natural action given by theaffine structure on G . The corresponding Lie algebra is the semi-direct product via the leftmultiplications given by the left-symmetric product. The two Lie group structure on T ∗ G defined above are not isomorphic.It is also well known, since the works of Drinfel’d ([32]), that T ∗ G (with the Lie groupstructure performed by semi-direct product G ⋉ G ∗ via the coadjoint representation of G on G ∗ ) is a particular case of the large class of the so-called Drinfel’d double Lie groups.As a Drinfel’d double Lie group, T ∗ G admits a metric which is invariant under left andright translations ([32]). This Lie group structure on T ∗ G does not admit a left-invariantsymplectic form, except in the Abelian case. In this thesis we deal with the Lie group structure which make T ∗ G aDrinfel’d double Lie group of G . Studying the geometry of a given Lie group is to study invariant structures on it. Thecotangent bundle T ∗ G of a Lie group G can exhibit very interesting and rich algebraic andgeometric structures (affine, symplectic, pseudo-Riemannian, Kählerian,...[56], [47], [34],[32], [28], [5]).Such structures can be better understood when one can exhibit the group of transfor-mations which preserve them. This very often involves the automorphisms of T ∗ G , if in On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 eneral Introduction particular, such structures are invariant under left or right multiplications by the elementsof T ∗ G . This is one of the reason for which we deal with automorphisms of cotangent bun-dles of Lie groups in Chapter II of this dissertation. We study the connected componentof the unit of the group of automorphisms of the Lie algebra D := T ∗ G of T ∗ G . Such aconnected component being spanned by exponentials of derivations of D , we often workwith those derivations. Let der ( G ) stand for the Lie algebra of derivations of G , while J denotes that of linear maps j : G → G satisfying j ([ x, y ]) = [ j ( x ) , y ] , for every elements x, y of G . We give a characterization of all derivations of T ∗ G (Theorem 2.3.1) and showthat in particular, if G has a bi-invariant Riemannian or pseudo-Riemannian metric, thenevery derivation φ of D can be expressed in terms of elements of der ( G ) and J alone. Fur-thermore, we give prominence to the fact that the Lie group Aut ( D ) of automorphismsof D is a super symmetric Lie group and its Lie algebra der ( D ) possesses Lie subalgebraswhich are Lie superalgebras, i.e. they are Z / Z -graded Lie algebras with the Lie bracketsatisfying [ x, y ] = − ( − deg ( x ) deg ( y ) [ y, x ] (Theorem 2.3.2). We also consider particular cases(e.g. orthogonal Lie algebras, semi-simple Lie algebras, compact Lie algebras) and recoverby simple methods interesting cohomological known results (Section 2.3.6).In Chapter III we completely characterise the space of prederivations of D , that isendomorphisms p of D which satisfy p (cid:0)(cid:2) x, [ y, z ] (cid:3)(cid:1) = (cid:2) p ( x ) , [ y, z ] (cid:3) + (cid:2) x, [ p ( y ) , z ] (cid:3) + (cid:2) x, [ y, p ( z )] (cid:3) , for every elements x, y, z of D . The Lie algebra der ( D ) is a subalgebra of the Lie algebraPder ( D ) of prederivations of D . Prederivations can be used to study bi-invariant metricson Lie groups ([6], [9], [64]). One of the important results within this chapter is: If G isa semi-simple Lie group with Lie algebra G , then any prederivations of T ∗ G (not semi-simple) is a derivation . This is an extension of the result of Müller ([64]) which states thatany prederivation of a semi-simple Lie algebra is a derivation, hence an inner derivation.We also give a structure theorem for Pder ( D ) which states that Pder ( D ) decomposes intoPder ( T ∗ G ) = G ⊕ G , where G is a reductive subalgebra of Pder ( D ) , that is [ G , G ] ⊂ G and [ G , G ] ⊂ G . Semi-simple, compact and more generally orthogonal Lie algebras arealso considered in this chapter.We study bi-invariant metrics of cotangent bundles of Lie groups in Chapter IV . Inthis chapter we characterise all orthogonal structures on T ∗ G (Theorem 4.3.1) and theirisometries. It is known that if ( G, µ ) is a connected and simply-connected orthogonal Liegroup with Lie algebra G , then the isotropy group of the neutral element of G in the group I ( G, µ ) of isometries of ( G, µ ) is isomorphic to the group of preautomorphisms of G whichpreserve the non-degenerate bilinear form induced by µ on G and whose Lie algebra is thewhole set of skew-symmetric prederivations of G ([64]). We characterise the isometries of bi-invariant metrics through the skew-symmetric prederivations with respect to the orthogonalstructures induced on T ∗ G by the bi-invariant metrics (Proposition 4.3.2). In the case where G possesses an orthogonal structure, we proved that any orthogonal structure on T ∗ G canbe expressed in terms of the duality pairing and endomorphisms of G which commute withall adjoint operators (Theorem 4.4.1). If G is a semi-simple Lie algebra, we prove that anyprederivation of T ∗ G which is skew-symmetric with respect to any orthogonal structure on On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 eneral Introduction T ∗ G is an inner derivation (Proposition 4.5.1); that is the connected component of the unitof the Lie group of isometries of any bi-invariant metric on T ∗ G is spanned by exponentialsof inner derivations of T ∗ G . Examples of the affine Lie group of the real line, the speciallinear group SL (2 , R ) , the group SO (3 , R ) of rotations and the -dimensional oscillatorgroup are given.The geometry of the Lie group of affine motions of the real line is explored in ChapterV . The geodesics of the left invariant affine structure induced by the symplectic structureare studied as well as integrale curves of left invariant vector fields. Since, the symplecticform considered on the affine Lie group corresponds to an invertible solution of the ClassicalYang-Baxter equation, we have also studied the geometry of the corresponding double Liegroup. The affine and complex structures on the double introduced by Diatta and Medina([28]) are considered.
On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 hapter One
Invariant Structures on Lie Groups
Contents
This chapter is to make this thesis as self contained as possible. It defines basic notionsand terminologies which might be useful throughout this dissertation.
Let G be a Lie group, ǫ its identity element, G its Lie algebra and T G its tangent bundle.Let G ∗ stand for the dual space of G and let K stand for the field R of real numbers or thefield C of complex numbers. Definition 1.1.1.
A bi-invariant pseudo-metric on G is a function µ : T G → K which isquadratic on each fiber, nondegenerate and invariant under both left and right translationsof the group G . A bi-invariant pseudo-metric on the Lie group G corresponds to a non-degeneratequadratic form q : G → K for which the adjoint operators are skew-symmetric; that is,if µ also denotes the corresponding symmetric bilinear form on G , we have µ ([ x, y ] , z ) + µ ( y, [ x, z ]) = 0 (1.1)for every x, y, z in G . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 rthogonal Structures on Lie Groups Conversely, if G admits a non-degenerate symmetric bilinear form for which the adjointoperators are skew-symmetric, then there exists, on every connected Lie group with Liealgebra G , a bi-invariant pseudo-metric.Throughout this work, a non-degenerate symmetric bilinear form on G for which theadjoint operators are skew-symmetric is called simply a bi-invariant scalar product or anorthogonal structure on G as well as a bi-invariant pseudo-metric on G is called a bi-invariant metric or an orthogonal structure on G .Lie groups (resp. Lie algebras) with orthogonal structures are called orthogonal orquadratic (see e.g. [58], [60]). In [3] orthogonal Lie algebras are called metrizable Lie alge-bras .In [58], Medina and Revoy have shown that every orthogonal Lie algebra is obtainedby the so-called double extension procedure.Consider the isomorphism of vector spaces θ : G → G ∗ defined by h θ ( x ) , y i := µ ( x, y ) , (1.2)where h , i on the left hand side, is the duality pairing h x, f i = f ( x ) , between elements x of G and f of G ∗ . Then, θ is an isomorphism of G -modules in the sense that it is equivariantwith respect to the adjoint and coadjoint actions of G on G and G ∗ respectively; i.e. θ ◦ ad x = ad ∗ x ◦ θ, (1.3)for all x in G . Which is also θ − ◦ ad ∗ x = ad x ◦ θ − , (1.4)for all x in G . The converse is also true. More precisely, a Lie group (resp. algebra) isorthogonal if and only if its adjoint and coadjoint representations are isomorphic. SeeTheorem 1.4. of [60]. Example 1.1.1.
Any Abelian Lie algebra with any scalar product is an orthogonal Liealgebra.
Example 1.1.2.
Semi-simple Lie algebras with the Killing forms are orthogonal Lie alge-bras.
Example 1.1.3 (Oscillator groups or diamond groups) . Let λ = ( λ , λ , . . . , λ n ) , where < λ ≤ λ ≤ · · · ≤ λ n are positive real numbers. On R n +2 ≡ R × R × C n , define theoperation ( t, s ; z , . . . , z n ) · ( t ′ , s ′ ; z ′ , . . . , z ′ n ) = (cid:16) t + t ′ , s + s ′ + 12 n X j =1 Im (¯ z j z ′ j e iλ j t ) ; z + z ′ e iλ t , . . . , z n + z ′ n e iλ n t (cid:17) , (1.5)where t, t ′ , s, s ′ are real numbers while z i , z ′ i , i = 1 , , . . . , n , are in C . Endowed with theoperation (1.5) and the adjacent manifold structure, R n +2 is a Lie group of dimension n + 2 . This Lie group is noted by G λ and is called Oscillator group in [12], [29], [35],
On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 rthogonal Structures on Lie Groups [68] (for dimension n = 1 ), twisted Heisenberg group in [73] and diamond group inother literature ([73]).The Lie algebra of G λ , called oscillator algebra of dimension n + 2 , is the vectorspace spanned by { e − , e , e , e , . . . , e n , ˇ e , ˇ e , . . . , ˇ e n } with the following non-zero brackets: [ e − , e j ] = λ j ˇ e j ; [ e − , ˇ e j ] = − λ j e j ; [ e j , ˇ e j ] = e . (1.6)The oscillator Lie algebra is noted by G λ . Every element x of G λ can be written: x = αe − + βe + n X j =1 α j e j + n X j =1 β j ˇ e j . (1.7)where α, β, α j , β j ( ≤ j ≤ n ) are real numbers. Then the following quadratic form definesan orthogonal structure on G λ (see [12]): k λ ( x, x ) := 2 αβ + n X j =1 λ j ( α j + β j ) (1.8)Oscillator groups are subject of a lot of studies ([29], [31], [35], [36], [50], [59], [68] ). Theyappear in many branches of Physics and Mathematical Physics and give particular solutionsof Einstein-Yang-Mills equations ([50]). Definition 1.1.2.
A hyperbolic plan E is a -dimensional linear space endowed with anon-degenerate symmetric bilinear form B such that there exists a non-zero element v of E with B ( v, v ) = 0 . A hyperbolic space is an orthogonal sum of hyperbolic plans. Remark 1.1.1.
A hyperbolic space is of even dimension.For more about hyperbolic spaces, see [48].
Definition 1.1.3.
A hyperbolic Lie algebra is an orthogonal Lie algebra which containstwo totally isotropic subspaces in duality for the orthogonal structure.
Definition 1.1.4.
A Manin Lie algebra is an orthogonal Lie algebra G such that: • G admits two totally isotropic subalgebras h and h ; • h and h are in duality one to the other for the orthogonal structure of G .In this case, ( G , h , h ) is called a Manin triple. On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 oisson Structure on Lie Groups Definition 1.2.1. A C ∞ -smooth Poisson structure (Poisson bracket) on a C ∞ -smoothfinite-dimensional manifold M is an R -bilinear skew-symmetric operation C ∞ ( M ) × C ∞ ( M ) −→ C ∞ ( M )( f, g ) f, g } (1.9) on the space C ∞ ( M ) of real-valued C ∞ -smooth functions on M , such that • (cid:0) C ∞ ( M ) , { , } (cid:1) is a Lie algebra; • { , } is a derivation in each factor, that is it verifies the Liebniz identity { f, gh } = { f, g } h + g { f, h } (1.10) for every f, g, h in C ∞ ( M ) .A manifold equipped with such a bracket is called a Poisson manifold . Example 1.2.1.
Any manifold carries a trivial Poisson structure. One just has to put { f, g } = 0 , for all smooth functions f and g on M . Example 1.2.2.
Let ( x, y ) denote coordinates on R and p : R → R be an arbitrarysmooth function. One defines a smooth Poisson structure on R by putting { f, g } = (cid:18) ∂f∂x ∂g∂y − ∂f∂y ∂g∂x (cid:19) p, (1.11)for every f, g in C ∞ ( R ) . Example 1.2.3.
A symplectic manifold ( M, ω ) is a manifold M equipped with a non-degenerate closed differential -form ω , called a symplectic form . If f : M → R is afunction on ( M, ω ) , we define its Hamiltonian vector field , denoted by X f , as follows: i X f ω := ω ( X f , · ) = − T f, (1.12)where
T f stands for the differential map of f . One defines on ( M, ω ) a natural bracket,called the Poisson bracket of ω , as follows: { f, g } = ω ( X f , X g ) = −h T f, X g i = − X g ( f ) = X f ( g ) , (1.13)for every f, g ∈ C ∞ ( M ) . Thus, any symplectic manifold carries a Poisson structure.Let ( M, { , } ) be a Poisson manifold. To every f in C ∞ ( M ) corresponds a unique vectorfield X f on M such that X f ( g ) = { f, g } , (1.14)for every g ∈ C ∞ ( M ) (see e.g. [57]). This is an extension of the notion of Hamiltonian vectorfield from the symplectic to the Poisson context. Thus, X f is called the Hamiltonianvector field associated to f as well as f is called the Hamiltonian function of X f . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 oisson Structure on Lie Groups
A map φ : ( M , { , } ) → ( M , { , } ) between two Poisson manifolds issaid to be a Poisson morphism if it is smooth and satisfies { f ◦ φ, g ◦ φ } = { f, g } ◦ φ, (1.15) for every f, g in C ∞ ( M ) . Example 1.2.4. If ( M , { , } ) and ( M , { , } ) are Poisson manifolds, then M × M has aPoisson structure characterized by the following properties:1. the projections π i : M × M → M i , i = 1 , , are Poisson morphisms;2. { f ◦ π , g ◦ π } = 0 , for any f in C ∞ ( M ) and g in C ∞ ( M ) . Example 1.2.5 (Kirilov-Kostant-Sauriau (KKS)) . Let G be a finite dimensional Lie alge-bra seen as the space of linear maps on its dual G ∗ ; i.e. G ≡ ( G ∗ ) ∗ ⊂ C ∞ ( G ∗ ) . Let f, g bein C ∞ ( G ∗ ) and α belongs to G ∗ . If T α f and T α g denote the differentials of f and g at thepoint α (seen as elements of G ), one defines a linear Poisson structure on G ∗ by putting: { f, g } ( α ) := h α, [ T α f, T α g ] G i , (1.16)where h , i in the right hand side stands for the pairing between G and its dual.This structure plays an important role in many domains of mathematical physics, quan-tization, hyper-kählerian geometry, ...Let M be a smooth manifold of dimension n ( n ∈ N ∗ ) and p a positive integer. Denoteby Λ p T M the space of tangent p -vectors of M . It is a vector bundle over M whose fiber overeach point x ∈ M is the space Λ p T x M = Λ p ( T x M ) , which is the exterior antisymmetricproduct of p copies of the tangent space T x M . Of course Λ T M = T M .Let ( x , . . . , x n ) be a local system of coordinates at x ∈ M . Then Λ p T x M admits alinear basis consisting of the elements ∂∂x i ∧ . . . ∧ ∂∂x ip ( x ) with i < i < . . . < i p . Definition 1.2.3.
A smooth p -vector field π on M is a smooth section of Λ p T M , i.e. amap π from M to Λ p T M , which associates to each point x of M a p -vector π ( x ) of Λ p T x M ,in a smooth way. Given a -vector field π on a smooth manifold M , one defines a tensor field, also denotedby π , by the following formula: { f, h } := π ( f, h ) := h T f ∧ T h, π i (1.17) Definition 1.2.4. A -vector field π , such that the bracket given by (1.17) is a Poissonbracket, is called a Poisson tensor, or also a Poisson structure. Any Poisson tensor π arises from a -vector field which we will also denote by π . Example 1.2.6.
The Poisson tensor corresponding to the standard symplectic structure ω = P nk =1 dx k ∧ dy k on R n is P nk =1 ∂∂x k ∧ ∂∂y k . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 oisson Structure on Lie Groups Definition 1.2.5.
Let G be a Lie group. A Poisson-Lie structure on G is a Poissontensor π on the underlying manifold of G such that the multiplication map G × G → G ( g, g ′ ) gg ′ (1.18) is a Poisson morphism (is grouped as said in [32]); G × G being equipped with the Poissontensor product π × π .A Lie group G with a Poisson-Lie structure π is called a Poisson-Lie group and isdenoted by ( G, π ) . The Definition 1.2.5 is equivalent to the following: π ( gh ) = T h L g · π ( h ) + T g R h · π ( g ) , (1.19)for every g, h in G , where the differentials T g L h and T g R h of the left translation L g andthe right translation R g are naturally extended to the linear space Λ T g G ; i.e. T g L h · ( X ∧ Y ) := ( T g L h · X ) ∧ ( T g L h · Y ) , (1.20) T g R h · ( X ∧ Y ) := ( T g R h · X ) ∧ ( T g R h · Y ) , (1.21)for any X, Y in T ∗ g G . Definition 1.2.6.
Let G be a Lie group. A tensor field π on G is called multiplicative if itsatisfies Relation (1.19). Example 1.2.7.
To every Lie algebra corresponds, at least, one Poisson-Lie group. Indeed,the dual space (seen as an Abelian Lie group) of any Lie algebra, endowed with its linearPoisson structure given in Example 1.2.5 is a Poisson-Lie group.Every Lie group possesses, at least, one non-trivial Poisson tensor (see [24]).
Let G be a Lie group with Lie algebra G , π a Poisson-Lie tensor on G with correspondingbracket { , } . One obtains a Lie algebra structure [ , ] ∗ on the dual space G ∗ of G by setting [ α, β ] ∗ := T ǫ { f, g } , (1.22)where f and g are smooth functions on G such that α and β are equal, respectively, to thedifferentials of f and g at the identity element ǫ of G : T ǫ f = α , T ǫ g = β . Definition 1.2.7.
1. The bracket (1.22) is called the ”linearized” bracket of π at ǫ andthe pair ( G ∗ , [ , ] ∗ ) is said to be the "linearized" or the dual Lie algebra of ( G, π ) orof ( G , λ ) , where λ := T ǫ π : G → Λ G ( λ is the transpose of [ , ] ∗ : Λ G ∗ → G ∗ ).2. Every Lie group, with Lie algebra ( G ∗ , [ , ] ∗ ) is called a dual Lie group of ( G, π ) . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 oisson Structure on Lie Groups
3. A subgroup H of a Poisson-Lie group ( G, π ) is said to be a Poisson-Lie subgroup of ( G, π ) if it is also a Poisson submanifold of ( G, π ) . Definition 1.2.8.
Let G be a Lie algebra.1. A Lie bi-algebra structure on G is a -cocycle λ : G → Λ G for the adjoint rep-resentation of G such that its transpose λ t : Λ G ∗ → G ∗ defines a Lie bracket on thevector space G ∗ .2. A Lie algebra, with a Lie bi-algebra structure, is called a Lie bi-algebra. A Lie bi-algebra will be denoted by ( G , λ ) or ( G , G ∗ ) . Theorem 1.2.1. ([32]) Let G be a simply connected Lie group. A Poisson structure, withPoisson tensor π , on G bijectively corresponds to a Lie bi-algebra structure λ = T ǫ π on theLie algebra G of G . Let ( G , [ , ]) be a Lie algebra. Suppose ( G , G ∗ ) is a Lie bi-algebra and let [ , ] ∗ denote theinduced Lie bracket on G ∗ . Then, the transpose of the Lie bracket of G is a -cocycle ofthe Lie algebra ( G ∗ , [ , ] ∗ ) . Hence, the Lie algebras ( G , [ , ]) and ( G , [ , ] ∗ ) act one to the otherby their respective coadjoint actions. The space G × G ∗ can be equipped with the scalarproduct: h ( x, f ) , ( y, g ) i := f ( y ) + g ( x ) , (1.23)for every x, y in G and every f, g in G ∗ .We have the following result due to Drinfel’d. Theorem 1.2.2. ([32]) The following are equivalent:1. ( G , G ∗ ) is a Lie bi-algebra;2. D := ( G × G ∗ , h , i ) is equipped with a unique structure [ , ] D of orthogonal Lie algebrasuch that:(a) G and G ∗ are Lie subalgebras of D ;(b) G and G ∗ are totally isotropic and in duality for the scalar product h , i .In this case, for every x in G and g in G ∗ , [ x, g ] D = ad ∗ x g − ad ∗ g x (1.24) Definition 1.2.9.
The Lie algebra D of the Theorem 1.2.2 is called the Drinfel’d doubleLie algebra of ( G, π ) or of ( G , λ := T ǫ π ) . Every Lie group, with Lie algebra D , is called a Drinfel’d double Lie group of ( G, π ) . The Lie bracket on D reads: [( x, f ) , ( y, g )] D = (cid:0) [ x, y ] + ad ∗ f y − ad ∗ g x, [ f, g ] ∗ + ad ∗ x g − ad ∗ y f (cid:1) (1.25)for every ( x, f ) and ( y, g ) in D .Since its introduction in ([32]), the notion of Drinfel’d double attracted manyresearchers ([2], [13], [18], [21],[22], [24],[28], [46]). On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ang-Baxter Equation Let M be a smooth manifold. For any integer p , let Ω p ( M ) stand for the space of smoothsections of Λ p T M and let Ω ∗ ( M ) be the direct sum of the spaces Ω p ( M ) , where • Ω p ( M ) = { } , if p < ; • Ω ( M ) = C ∞ ( M ) ; • Ω ( M ) = X ( M ) (smooth vector fields on M ) ; • Ω p ( M ) = { } , if p > dim M . Theorem 1.3.1 (Schouten Bracket Theorem) . There exists a unique bilinear operation [ , ] : Ω ∗ ( M ) × Ω ∗ ( M ) → Ω ∗ ( M ) natural with respect to restriction to open sets, called the Schouten Bracket , that satisfies the following properties:1. [ , ] is a biderivation of degree − , i.e. for all homogeneous elements A and B of Ω ∗ ( M ) and any C in Ω ∗ ( M ) , • deg[ A, B ] = deg A + deg B − ; and • [ A, B ∧ C ] = [ A, B ] ∧ C + ( − (deg A +1) deg B B ∧ [ A, C ] ;2. [ , ] is defined on C ∞ ( M ) and on X ( M ) by(a) [ f, g ] = 0 , for any f, g in C ∞ ( M ) ;(b) [ X, f ] = X · f , for all f in C ∞ ( M ) and any vector field X on M ;(c) [ X, Y ] is the usual Jacobi-Lie bracket of vector fields if X and Y are in X ( M ) ;3. [ A, B ] = ( − deg A deg B [ B, A ] .In addition, we have the graded Jacobi identity ( − deg A deg C [[ A, B ] , C ] + ( − deg B deg A [[ B, C ] , A ] + ( − deg C deg B [[ C, A ] , B ] = 0 (1.26) for all homogeneous elements A, B, C of Ω ∗ ( M ) . Let G be a Lie group with Lie algebra G . On the Z -graded vector space Λ G := ⊕ p ∈ Z Λ p G we consider the structure of graded Lie algebra obtained by the extension of the Lie bracketof G satisfying the properties of the definition of the Schouten’s bracket.Let r ∈ Λ G and note by η the -coboundary defined by η ( g ) = Ad g r − r, (1.27)for all g in G .Let r + and r − denote the left invariant and right invariant tensor fields associated to r respectively. One wonders whether the corresponding multiplicative tensor π := r + − r − is Poisson. The answer is given by the On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ymmetric Spaces ([51]) Let Λ be a contravariant skew-symmetric -vector field on amanifold M .(i) Λ is Poisson if and only if [Λ , Λ] = 0 (this is equivalent to the Jacobi identity);(ii) If [Λ , Λ] = 0 , then the map ∂ : Ω ∗ ( M ) → Ω ∗ ( M ) P [Λ , P ] (1.28) is an operator of cohomology, i.e. ∂ Λ ◦ ∂ Λ = 0 . Definition 1.3.1.
The cohomology defined by ( ii ) of Proposition 1.3.1 is called the Poissoncohomology of the Poisson manifold ( M, Λ) . According to the Proposition 1.3.1, the tensor field π is Poisson if and only if the -tensor field [ π, π ] = [ r + , r + ] + [ r − , r − ] = [ r, r ] + − [ r, r ] − (1.29)vanishes identically or equivalently, for every g ∈ G , the -tensor [ π, π ] g := T ǫ ( Ad g [ r, r ] − [ r, r ]) (1.30)equals zero. Hence, π defines a Lie-Poisson tensor if and only if the -vector [ r, r ] is Ad G -invariant, i.e. for all g ∈ G , Ad g [ r, r ] = [ r, r ] (1.31) Definition 1.3.2.
1. Equation (1.31) below is called the
Generalized Yang-BaxterEquation (GYBE) and its solutions are called r -matrices .2. In the particular case where [ r, r ] = 0 , one says that r is a solution of the ClassicalYang-Baxter Equation (CYBE) . Let G be a connected Lie group with Lie algebra G and identity element ǫ . Definition 1.4.1. A symmetric space for G is a homogeneous space M ≡ G/H suchthat the isotropy group H of any arbitrary point is an open subgroup of the fixed point set G σ := { g ∈ G : σ ( g ) = g } of an involution σ of G . The involution σ is in fact an automorphism of G and fixes the identity element ǫ .Hence, the differential at ǫ of σ is an automorphism, also denote by σ , of the Lie algebra G with square equal to the identity mapping of G : σ = Id G . Then the eigenvalues of σ are +1 and − . The eigenspace associated to +1 is the Lie algebra h of H . We denote the − eigenspace by m . Since σ is a automorphism of G , we have the following decomposition G = h ⊕ m (1.32)with [ h , h ] ⊂ h ; [ h , m ] ⊂ m ; [ m , m ] ⊂ h . (1.33)Conversely, given any Lie algebra G with direct sum decomposition (1.32) satisfying (1.33),the linear map σ , equal to the identity on h and minus the identity on m , is an involutiveautomorphism of G . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ie Superalgebras Let G be a finite dimensional Lie algebra over a field K of characteristic zero. Definition 1.5.1.
A subalgebra h is said to be reductive in G if G is a semisimple h -module;that is G is a sum of simple h -modules in the adjoint representation of h in G . The Liealgebra G is said to be reductive if it is a reductive subalgebra of itself. Let G be an arbitrary finite dimensional Lie algebra over K , V be a semisimple G -moduleand I be the ideal I = { x ∈ G : x · V = { }} . Now set h = G /I . By a result (see [16]) dueto E. Cartan and N. Jacobson, we have h = [ h , h ] + Z ( h ) , where Z ( h ) is the center of h , [ h , h ] is semisimple, and V is semisimple as a Z ( h ) -module.Now suppose that G is reductive. Since the center Z ( G ) of G is a G -submodule of G ,there is an ideal J of G such that G is the direct sum J ⊕ Z ( G ) . By the result we have citedabove, we have J = [ J, J ] + Z ( J ) , where Z ( J ) is the center of J and [ J, J ] is semisimple.But, of course, [ J, J ] = [ G , G ] and Z ( J ) = { } . Hence, G = [ G , G ] + Z ( G ) and [ G , G ] issemisimple. Conversely, it is clear that if a finite dimensional Lie algebra G over K satisfiesthese last conditions, then G is reductive. Let K be an Abelian field of characteristic zero. Definition 1.6.1. A K -linear space L is a K -superalgebra, with superbracket [ · , · ] , if a ) it is Z / Z -graded, i.e. L = L ¯0 ⊕ L ¯1 ; [ L ¯0 , L ¯0 ] ⊂ L ¯0 ; [ L ¯0 , L ¯1 ] ⊂ L ¯1 ; [ L ¯1 , L ¯1 ] ⊂ L ¯0 (1.34) b ) for every homogeneous elements a, b, c of L , ( − | a |·| c | [ a, [ b, c ]] + ( − | b |·| a | [ b, [ c, a ]] + ( − | c |·| b | [ c, [ a, b ]] = 0 . (1.35) where | a | (respectively | b | and | c | ) stands for the degree of a (respectively b and c ). For a K -superalgebra L = L ¯0 ⊕ L ¯1 , L ¯0 is an ordinary Lie algebra while L ¯1 is a L ¯0 -module.Each element z of L can be uniquely written: z = z ¯0 + z ¯1 (1.36)where z ¯0 ∈ L ¯0 and z ¯1 ∈ L ¯1 . One says that z ¯0 is the component of degree ¯0 of z and z ¯1 isthe component of degree ¯1 of z . On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ie Superalgebras See [42] for more about Lie superalgebras.
Definition 1.6.2.
Let L be a Lie superalgebra over K .1. A derivation of degree ¯0 of L is an endomorphism D : L → L such that D ( L ¯0 ) ⊂ L ¯0 ; D ( L ¯1 ) ⊂ L ¯1 ; D [ a, b ] = [ D ( a ) , b ] + [ a, D ( b )] (1.37) for every a, b ∈ L .2. A derivation of degree ¯1 of L is an endomorphism D : L → L such that D ( L ¯0 ) ⊂ L ¯1 ; D ( L ¯1 ) ⊂ L ¯0 ; D [ a, b ] = [ D ( a ) , b ] + ( − | a | [ a, D ( b )] (1.38) for every homogeneous element a of L and every element b ∈ L .3. More generaly, a derivation of degree r ( r ∈ Z := Z / Z ) of a Lie superalgebra L isan endomorphism D ∈ End r ( L ) := { ϕ ∈ End ( L ) : ϕ ( L s ) ⊂ L r + s } . Note by der ¯0 ( L ) the set of derivations of L of degree ¯0 and by der ¯1 ( L ) the set ofderivations of L of degree ¯1 . Then, der ( L ) = der ¯0 ( L ) ⊕ der ¯1 ( L ) (1.39)Hence, to know the derivations of a Lie superalgebra L it sufficies to know the derivationsof degree ¯0 and the derivations of degree ¯1 . If d is in der ( L ) , we write d = d ¯0 + d ¯1 (1.40)with d ¯0 ∈ der ¯0 ( L ) and d ¯1 ∈ der ¯1 ( L ) . Proposition 1.6.1.
Let d = d ¯0 + d ¯1 ∈ der ( L ) . Then,1. About d ¯0 :a) d ¯0 | L ¯0 : L ¯0 → L ¯0 is a derivation of the Lie algebra L ¯0 .b) If z ¯0 ∈ L ¯0 and z ¯1 ∈ L ¯1 , then d ¯0 [ z ¯0 , z ¯1 ] = [ d ¯0 ( z ¯0 ) , z ¯1 ] + [ z ¯0 , d ¯0 ( z ¯1 )]; (1.41) that is the endomorphism d ¯0 | L ¯1 : L ¯1 → L ¯1 verifies [ d ¯0 , ad z ¯0 ] | L ¯1 = ad ( d ¯0 ( z ¯0 )) | L ¯1 . (1.42) c) For every z ¯1 , z ′ ¯1 ∈ L ¯1 , d ¯0 [ z ¯1 , z ′ ¯1 ] = [ d ¯0 ( z ¯1 ) , z ′ ¯1 ] + [ z ¯1 , d ¯0 ( z ′ ¯1 )] (1.43)
2. About d ¯1 :a) The morphism d ¯1 | L ¯0 : L ¯0 → L ¯1 verifies d ¯1 [ z ¯0 , z ′ ¯0 ] = [ d ¯1 ( z ¯0 ) , z ′ ¯0 ] + [ z ¯0 , d ¯1 ( z ′ ¯0 )] , (1.44) for every z ¯0 , z ′ ¯0 ∈ L ¯0 ; i.e. d ¯1 | L ¯0 : L ¯0 → L ¯1 is a -cocycle.b) The morphism d ¯1 | L ¯1 : L ¯1 → L ¯0 satisfies d ¯1 [ z ¯0 , z ¯1 ] = [ d ¯1 ( z ¯0 ) , z ¯1 ] + [ z ¯0 , d ¯1 ( z ¯1 )] (1.45) for every z ¯0 ∈ L ¯0 and every z ¯1 ∈ L ¯1 On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ohomology of Lie Algebras Let G be a Lie algebra over a field K of characteristic zero. Definition 1.7.1. A G -module is a linear space V of same dimension than G with a bilinearmap ϕ : G × V → V such that ϕ ([ x, y ] , v ) = ϕ ( x, ϕ ( y, v )) − ϕ ( y, ϕ ( x, v )) , (1.46) for every elements x , y of G and every vector v in V . A G -module corresponds to a representation of G on the linear space V , that is ahomomorphism Φ : G → gl ( V ) defined by: Φ( x )( v ) = ϕ ( x, v ) := x · v, (1.47)for every x in G and every v in V . Definition 1.7.2.
Let V be a G -module and p be a non-zero integer ( p ∈ N ∗ ). • A cochain of G of degree p (or a p -cochain) with values in V is a skew-symmetric p -linear map from g p = p times z }| { g × · · · × g to V . • A cochain of degree (or a -cochain) of G with values in V is a constant map from G to V . Note by C p ( G , V ) the space of p -cochains of G with values in V . We have: C p ( G , V ) = Hom (Λ p G , V ) , si p ≥ V si p = 0; { } si p < . (1.48)The space of cochains of G with values in V is denoted by C ∗ ( G , V ) := ⊕ p C p ( G , V ) . Oneendows C p ( G , V ) with a G -module structure by setting : ( x · Φ)( x , . . . , x p ) = x · Φ( x , . . . , x p ) − X ≤ i ≤ p Φ( x , . . . , x i − , [ x, x i ] , x i +1 , . . . , x p ) (1.49)for all x, x , . . . , x p in G and all Φ in C p ( G , V ) . This structure can be extended to the space C ∗ ( G , V ) .Let us now define the endomorphism d : C ∗ ( G , V ) → C ∗ ( G , V ) , called coboundaryoperator: • If Φ is in C ( G , V ) = V and x is in G , then ( d Φ)( x ) = d Φ( x ) = x · Φ . (1.50) On the Geometry of Cotangent Bundles of Lie Groups
Bakary MANGA c (cid:13)
URMPM/IMSP 2010 ffine Lie Groups • For p ≥ , x , . . . , x p +1 in g and Φ in C p ( g , V ) , ( d Φ)( x , . . . , x p +1 ) = X ≤ s ≤ p +1 ( − s +1 x s · Φ( x , . . . , ˆ x s , . . . , x p +1 ) (1.51) + X ≤ s An element of Z p ( G , V ) is called a cocyle of degree p (or p -cocycle) of G with values in V while an element of B p ( G , V ) is said to be a coboundary of degree p (or p -coboundary) of g with values in V . Since d = 0 , one has: B p ( G , V ) ⊂ Z p ( G , V ) . Then, we set, for p ≥ , H p ( G , V ) := Z p ( G , V ) /B p ( G , V ) . (1.52) Definition 1.7.4. H p ( G , V ) is the space of cohomology of G of degree p (or p th space ofcohomology of G ) with values in V . Definition 1.8.1. A n -dimensional affine manifold is smooth manifold M equipped with asmooth atlas ( U i , Φ i ) i such that the transition functions Φ i ◦ Φ − j | Φ j ( U i ∩ U j ) : Φ j ( U i ∩ U j ) → Φ i ( U i ∩ U j ) , whenever they exist, are restrictions of affine transformations of R n . The definition below is equivalent to the following: if ( U i , Φ i ) and ( U j , Φ j ) are twocharts of the atlas satisfy U i ∩ U j then there exists an element θ ij of the affine groupAff ( R n ) = GL ( n, R ) ⋉ R n of R n such that Φ i ◦ Φ − j | Φ j ( U i ∩ U j ) = θ ij | Φ j ( U i ∩ U j ) . (1.53)Recall that Aff ( R n ) is the group of diffeomorphimsms of R n which preserve the standardconnection ∇ of R n : ∇ X Y := n X k =1 ( X · f k ) ∂∂x k , (1.54)where Y = P nk =1 f k ∂∂x k . Every open set U i is then endowed with a connection ∇ i whichis the reciprocal image by Φ i of the connection induced on Φ i ( U i ) by ∇ . The connection ∇ i is torsion free and without curvature. Since the transition function preserve ∇ , theconnections ∇ i can be perfectly glued into a zero-curvature and torsion free connection onthe manifold M . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ffine Lie Groups An application f : M → N between two affine manifold is called anaffine transformation if • it is smooth ; • for all charts ( U, Φ) of M and ( V, Ψ) of N such that f − ( V ) ∩ U = ∅ , the function Ψ ◦ f ◦ Φ − : Φ( f − ( V ) ∩ U ) → Ψ( f − ( V ) ∩ U ) is the restriction of an element of Aff ( R n ) . Definition 1.8.3. A Lie group G endowed with a left-invariant affine structure is calledan affine Lie group. In other words, an affine Lie group is a Lie group equipped with anaffine structure for which the left translations are affine transformations. The existence of affine structures is a difficult and interesting problem ([39],[63], [54], [52]). On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 hapter Two Automorphisms of CotangentBundles of Lie Groups Contents D := T ∗ G . . . . . . . . . . . . . . 232.4 Case of Orthogonal Lie Algebras . . . . . . . . . . . . . . . . . 362.5 Some Possible Applications and Open Problems . . . . . . . . 43 Let G be a Lie group whose Lie algebra G is identified with its tangent space T ǫ G at theunit ǫ . Throughout this work the cotangent bundle T ∗ G of G , is seen as a Lie group whichis obtained by the semi-direct product G ⋉ G ∗ of G and the Abelian Lie group G ∗ , where G acts on the dual space G ∗ of G via the coadjoint action. Here, using the trivialization byleft translations, the manifold underlying T ∗ G has been identified with the trivial bundle G × G ∗ . We sometimes refer to the above Lie group structure on T ∗ G, as its natural Liegroup structure. The Lie algebra Lie ( T ∗ G ) = G ⋉ G ∗ of T ∗ G will be denoted by T ∗ G orsimply by D .It is our aim in this work to fully study the connected component of the unit of thegroup of automorphisms of the Lie algebra D := T ∗ G . Such a connected component beingspanned by exponentials of derivations of D , we will work with those derivations andthe first cohomology space H ( D , D ) , where D is seen as the D -module for the adjointrepresentation.Our motivation for this work comes from several interesting algebraic and geometricproblems.The cotangent bundle T ∗ G of a Lie group G can exhibit very interesting and richalgebraic and geometric structures ([5], [28], [32], [34], [47], [56]). Such structures can bebetter understood when one can compare, deform or classify them. This very often involves On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ntroduction the invertible homomorphisms (automorphisms) of T ∗ G , if in particular, such structuresare invariant under left or right multiplications by the elements of T ∗ G . The derivatives atthe unit of automorphisms of the Lie group T ∗ G are automorphisms of the Lie algebra D . Conversely, if G is connected and simply connected, then so is T ∗ G and every automorphismof the Lie algebra D integrates to an automorphism of the Lie group T ∗ G . A probleminvolving left or right invariant structures on a Lie group also usually transfers to one onits Lie algebra, with the Lie algebra automorphisms used as a means to compare or classifythe corresponding induced structures.In the purely algebraic point of view, finding and understanding the derivations of agiven Lie algebra, is in itself an interesting problem ([23], [30], [40], [41], [49], [69]).On the other hand, as a Lie group, the cotangent bundle T ∗ G is a common Drinfel’ddouble Lie group for all exact Poisson-Lie structures given by solutions of the ClassicalYang-Baxter Equation in G . See e.g. [27]. Double Lie algebras (resp. groups) encode in-formation on integrable Hamiltonian systems and Lax pairs ([4], [10], [32], [55]), Poissonhomogeneous spaces of Poisson-Lie groups and the symplectic foliation of the correspond-ing Poisson structures ([27], [32], [55]). To that extend, the complete description of thegroup of automorphisms of the double Lie algebra of a Poisson-Lie structure would be abig contribution towards solving very interesting and hard problems. See Section 2.5 forwider discussions.Interestingly, the space of derivations of D encompasses interesting spaces of operatorson G , among which the derivations of G , the second space of the left invariant de Rhamcohomology H inv ( G, R ) of G , bi-invariant endomorphisms, in particular operators givingrise to complex group structure in G , when they exist.Throughout this work, der ( G ) will stand for the Lie algebra of derivations of G , while J will denote that of linear maps j : G → G satisfying j ([ x, y ]) = [ j ( x ) , y ] , for every elements x, y of G . We consider Lie groups and Lie algebras over the field R . However, most of theresults within this chapter are valid for any field of characteristic zero.We summarize some of our main results as follows. Theorem A. Let G be a Lie group, G its Lie algebra, T ∗ G its cotangent bundle and D := G ⋉ G ∗ = Lie ( T ∗ G ) . A derivation of D , has the form φ ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + f ◦ ( j − α ) (cid:17) , for all ( x, f ) in D ; where • α : G → G is a derivation of the Lie algebra G ; • the linear map j : G → G is in J ; • β : G → G ∗ is a -cocycle of G with values in G ∗ for the coadjoint action of G on G ∗ ; • ψ : G ∗ → G is a linear map satisfying the following conditions: for all x in G and all f, g in G ∗ , ψ ◦ ad ∗ x = ad x ◦ ψ and ad ∗ ψ ( f ) g = ad ∗ ψ ( g ) f. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ntroduction If G has a bi-invariant Riemannian or pseudo-Riemannian metric, say µ, then everyderivation φ of D can be expressed in terms of elements of der ( G ) and J alone, as follows, φ ( x, f ) = (cid:16) α ( x ) + j ◦ θ − ( f ) , θ ◦ α ′ ( x ) + f ◦ ( j ′ − α ) (cid:17) , for any element ( x, f ) of D , where α, α ′ are derivations of G ; the maps j, j ′ are in J and θ : G → G ∗ with θ ( x )( y ) := µ ( x, y ) , for every elements x, y of G . Theorem B. Let G be a Lie group and G its Lie algebra. The group Aut( D ) of auto-morphisms of the Lie algebra D of the cotangent bundle T ∗ G of G , is a super symmet-ric Lie group. More precisely, its Lie algebra der ( D ) is a Z / Z -graded symmetric (super-symmetric) Lie algebra which decomposes into a direct sum of vector spaces der ( D ) := G ⊕ G , with [ G i , G i ′ ] ⊂ G i + i ′ , i, i ′ ∈ Z / Z = { , } where G is the Lie algebra G := n φ : D → D , φ ( x, f ) = (cid:16) α ( x ) , f ◦ ( α − j ) (cid:17) , with α ∈ der ( G ) and j ∈ J o and G is the direct sum (as a vector space) of the space Q of 1-cocycles β : G → G ∗ andthe space Ψ of equivariant maps ψ : G ∗ → G with respect to the coadjoint and the adjointrepresentations, satisfying ad ∗ ψ ( f ) g = ad ∗ ψ ( g ) f, for every elements f, g of G ∗ .Moreover, G ⊕ ˜ G and G ⊕ ˜ G ′ are subalgebras of der ( D ) which are Lie superalgebras,i.e. they are Z / Z -graded Lie algebras with the Lie bracket satisfying [ x, y ] = − ( − deg ( x ) deg ( y ) [ y, x ] , where ˜ G := Q and ˜ G ′ := Ψ are Abelian subalgebras of der ( D ) and deg ( x ) = i , if x ∈ G i . The Lie superalgebras G ⊕ ˜ G and G ⊕ ˜ G ′ respectively correspond to the subalgebrasof all elements of der( D ) which preserve the subalgebra G and the ideal G ∗ of D . Theorem C. The first cohomology space H ( D , D ) of the (Chevalley-Eilenberg) coho-mology associated with the adjoint action of D on itself, satisfies H ( D , D ) ∼ = H ( G , G ) ⊕ J t ⊕ H ( G , G ∗ ) ⊕ Ψ , where H ( G , G ) and H ( G , G ∗ ) are the first cohomology spaces associated with the adjointand coadjoint actions of G , respectively; and J t := { j t , j ∈ J } (space of transposes ofelements of J ).If G is semi-simple, then Ψ = { } and thus H ( D , D ) ∼ = J t . Moreover, we have J ∼ = R p ,where p is the number of the simple ideals s i of G such that G = s ⊕ · · · ⊕ s p . Hence, ofcourse, H ( D , D ) ∼ = R p . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reliminaries If G is a compact Lie algebra, with centre Z ( G ) , we get H ( G , G ) ∼ = End ( Z ( G )) , J ∼ = R p ⊕ End ( Z ( G )) ,H ( G , G ∗ ) ∼ = L ( Z ( G ) , Z ( G ) ∗ ) , Ψ ∼ = L ( Z ( G ) ∗ , Z ( G )) . Hence, we get H ( D , D ) ∼ = ( End ( R k )) ⊕ R p , where k is the dimension of the center of G , and p is the number of the simple componentsof the derived ideal [ G , G ] of G . Here, if E, F are vector spaces, L ( E, F ) is the space oflinear maps E → F. Traditionally, spectral sequences are used as a powerful tool for the study of the co-homology spaces of extensions of Lie groups or Lie algebras and more generally, of locallytrivial fiber bundles (see e.g. [44], [65] for very interesting results and discussions). However,for the purpose of this investigation, we use a direct approach. Some parts of Theorem Ccan also be seen as a refinement, using our direct approach, of some already known results([44]). Its proof is given by different lemmas and propositions, discussed in Sections 2.3.6and 2.4.The chapter is organized as follows. In Section 2.2, we explain some of the materialand terminology needed to make the chapter more self contained. Sections 2.3 and 2.4 arethe actual core of the work where the main calculations and proofs of theorems are carriedout. In Section 2.5, we discuss some subjects related to this work, as well as some of thepossible applications. Although not central to the main purpose of the work within this chapter, the followingmaterial might be useful, at least, as regards parts of the terminology used throughout thischapter. Throughout this chapter, given a Lie group G , we will always let G ⋉ G ∗ stand for the Liegroup consisting of the Cartesian product G × G ∗ as its underlying manifold, together withthe group structure obtained by semi-direct product using the coadjoint action of G on G ∗ . Recall that the trivialization by left translations, or simply the left trivialization of T ∗ G isgiven by the following isomorphism ζ of vector bundles ζ : T ∗ G → G × G ∗ , ( σ, ν σ ) ( σ, ν σ ◦ T ǫ L σ ) , where L σ is the left multiplication L σ : G → G, τ L σ ( τ ) := στ by σ in G and T ǫ L σ is the derivative of L σ at the unit ǫ. In this chapter, T ∗ G will always be endowed with the On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reliminaries Lie group structure such that ζ is an isomorphism of Lie groups. The Lie algebra of T ∗ G is then the semi-direct product D := G ⋉ G ∗ . More precisely, the Lie bracket on D reads [( x, f ) , ( y, g )] := ([ x, y ] , ad ∗ x g − ad ∗ y f ) , (2.1)for any two elements ( x, f ) and ( y, g ) of D .In this work, we will refer to an object which is invariant under both left and righttranslations in a Lie group G , as a bi-invariant object. We discuss in this section, how T ∗ G is naturally endowed with a bi-invariant pseudo-Riemannian metric.The cotangent bundle of any Lie group (with its natural Lie group structure, as above)and in general any element of the larger and interesting family of the so-called Drinfel’ddoubles (see Section 2.2.2), are orthogonal Lie groups ([32]), as explained below.As above, let D := G ⋉ G ∗ be the Lie algebra of the cotangent bundle T ∗ G of G , seenas the semi-direct product of G by G ∗ via the coadjoint action of G on G ∗ , as in (2.1). Let µ stand for the duality pairing h , i , that is, for all ( x, f ) , ( y, g ) in D , µ (cid:16) ( x, f ) , ( y, g ) (cid:17) = f ( y ) + g ( x ) . (2.2)Then, µ satisfies the property (1.1) on D and hence gives rise to a bi-invariant (pseudo-Riemannian) metric on T ∗ G . We explain in this section how cotangent bundles of Lie groups are part of the broaderfamily of the so-called double Lie groups of Poisson-Lie groups.A Poisson structure on a manifold M is given by a Lie bracket { , } on the space C ∞ ( M, R ) of smooth real-valued functions on M, such that, for each f in C ∞ ( M, R ) , thelinear operator X f := { f, . } on C ∞ ( M, R ) , defined by g X f · g := { f, g } , is a vectorfield on M . The bracket { , } defines a -tensor, that is, a bivector field π which, seenas a bilinear skew-symmetric ’form’ on the space of differential -forms on M , is givenby π ( df, dg ) := { f, g } . The Jacobi identity for { , } now reads [ π, π ] S = 0 , where [ , ] S isthe so-called Schouten bracket, which is a natural extension to all multi-vector fields, ofthe natural Lie bracket of vector fields. Reciprocally, any bivector field π on M satisfying [ π, π ] S = 0 , is a Poisson tensor, i.e. defines a Poisson structure on M . See e.g. [55].Recall that a Poisson-Lie structure on a Lie group G, is given by a Poisson tensor π on G, such that, when the Cartesian product G × G is equipped with the Poisson tensor π × π , the multiplication m : ( σ, τ ) στ is a Poisson map between the Poisson manifolds ( G × G, π × π ) and ( G, π ) . In other words, the derivative m ∗ of m satisfies m ∗ ( π × π ) = π. If f, g are in G ∗ and ¯ f , ¯ g are C ∞ functions on G with respective derivatives f = ¯ f ∗ ,ǫ , g = ¯ g ∗ ,ǫ at the unit ǫ of G, one defines another element [ f, g ] ∗ of G ∗ by setting [ f, g ] ∗ := ( { ¯ f , ¯ g } ) ∗ ,ǫ . Then [ f, g ] ∗ does not depend on the choice of ¯ f and ¯ g as above, and ( G ∗ , [ , ] ∗ ) is a Liealgebra. Now, there is a symmetric role played by the spaces G and G ∗ , dual to each other.Indeed, as well as acting on G ∗ via the coadjoint action, G is also acted on by G ∗ using the On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G coadjoint action of ( G ∗ , [ , ] ∗ ) . A lot of the most interesting properties and applications of π, are encoded in the new Lie algebra ( G ⊕ G ∗ , [ , ] π ) , where [( x, f ) , ( y, g )] π := ([ x, y ] + ad ∗ f y − ad ∗ g x, ad ∗ x g − ad ∗ y f + [ f, g ] ∗ ) , (2.3)for every x, y in G and every f, g in G ∗ . The Lie algebras ( G ⊕ G ∗ , [ , ] π ) and ( G ∗ , [ , ] ∗ ) are respectively called the double andthe dual Lie algebras of the Poisson-Lie group ( G, π ) . Endowed with the duality pairingdefined in (2.2), the double Lie algebra of any Poisson-Lie group ( G, π ), is an orthogonalLie algebra, such that G and G ∗ are maximal totally isotropic (Lagrangian) subalgebras.The collection ( G ⊕ G ∗ , G , G ∗ ) is then called a Manin triple. More generally, ( G ⊕ G , G , G ) is called a Manin triple and ( G , G ) a bi-algebra or a Manin pair, if G ⊕ G is an orthogonal n -dimensional Lie algebra whose underlying adjoint-invariant pseudo-Riemannian metricis of index ( n, n ) and G , G are two Lagrangian complementary subalgebras. See [27],[32],[55] for wider discussions.Let r be an element of the wedge product ∧ G . Denote by r + (resp. r − ) the left (resp.right) invariant bivector field on G with value r = r + ǫ (resp. r = r − ǫ ) at ǫ . If π r := r + − r − is a Poisson tensor, then it is a Poisson-Lie tensor and r is called a solution of the Yang-Baxter Equation. If, in particular, r + is a (left invariant) Poisson tensor on G , then r is called a solution of the Classical Yang-Baxter Equation (CYBE) on G (or G ). In thislatter case, the double Lie algebra ( G ⊕ G ∗ , [ , ] π r ) is isomorphic to the Lie algebra D of thecotangent bundle T ∗ G of G . See e.g. [28]. We may also consider the linear map ˜ r : G ∗ → G , where ˜ r ( f ) := r ( f, . ) . The linear map θ r : ( G ⊕ G ∗ , [ , ] π r ) → D , θ r ( x, f ) := ( x + ˜ r ( f ) , f ) , isan isomorphism of Lie algebras, between D and the double Lie algebra of any Poisson-Liegroup structure on G, given by a solution r of the CYBE. D := T ∗ G D := T ∗ G Consider a Lie group G of dimension n , with Lie algebra G . Let us also denote by D thevector space underlying the Lie algebra D of the cotangent bundle T ∗ G , regarded as a D -module under the adjoint action of D . Consider the following complex with the coboundaryoperator ∂ , where ∂ ◦ ∂ = 0 : → D → Hom ( D , D ) → Hom (Λ D , D ) → · · · → Hom (Λ n D , D ) → . (2.4)We are interested in Hom ( D , D ) := { φ : D → D , φ linear } . The coboundary ∂φ oftheelement φ of Hom ( D , D ) is the element of Hom (Λ D , D ) defined by ∂φ ( u, v ) := ad u (cid:0) φ ( v ) (cid:1) − ad v (cid:0) φ ( u ) (cid:1) − φ ([ u, v ]) , (2.5)for any elements u = ( x, f ) and v = ( y, g ) in D . An element φ of Hom ( D , D ) is a -cocycleif ∂φ = 0 , i.e. φ ([ u, v ]) = ad u (cid:16) φ ( v ) (cid:17) − ad v (cid:16) φ ( u ) (cid:17) , = [ u, φ ( v )] + [ φ ( u ) , v ] . (2.6) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G In other words, -cocycles are the derivations of the Lie algebra D . In Section 2.3.6, wewill characterize the first cohomology space H ( D , D ) := ker( ∂ ) /Im ( ∂ ) of the associatedChevalley-Eilenberg cohomology, where for clarity, we have denoted by ∂ and ∂ thefollowing restrictions ∂ : D → Hom ( D , D ) and ∂ : Hom ( D , D ) → Hom ( ∧ D , D ) of thecoboudary operator ∂. Theorem 2.3.1. Let G be a Lie group, G its Lie algebra, T ∗ G its cotangent bundle and D := G ⋉ G ∗ the Lie algebra of T ∗ G . A -cocycle (for the adjoint representation) hence aderivation of D has the following form: φ ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ξ ( f ) (cid:17) , (2.7) for any ( x, f ) in D ; where- α : G → G is a derivation of the Lie algebra G ,- β : G → G ∗ is a -cocycle of G with values in G ∗ for the coadjoint action of G on G ∗ ,- ξ : G ∗ → G ∗ and ψ : G ∗ → G are linear maps satisfying the following conditions: [ ξ, ad ∗ x ] = ad ∗ α ( x ) , ∀ x ∈ G , (2.8) ψ ◦ ad ∗ x = ad x ◦ ψ, ∀ x ∈ G , (2.9) ad ∗ ψ ( f ) g = ad ∗ ψ ( g ) f, ∀ f, g ∈ G ∗ . (2.10)The rest of this section is dedicated to the proof of Theorem 2.3.1.Aiming to get a simpler expression for the derivations, let us write φ in terms of itscomponents relative to the decomposition of D into a direct sum D = G ⊕ G ∗ of vectorspaces as follows: for all ( x, f ) in D , φ ( x, f ) = (cid:16) φ ( x ) + φ ( f ) , φ ( x ) + φ ( f ) (cid:17) , (2.11)where φ : G → G , φ : G → G ∗ , φ : G ∗ → G and φ : G ∗ → G ∗ are all linear maps. In(2.11) we have made the identifications: x = ( x, , f = (0 , f ) so that the element ( x, f ) can also be written x + f . Likewise, we can write φ ( x ) = ( φ ( x ) , φ ( x )) ; φ ( f ) = ( φ ( f ) , φ ( f )) , (2.12)for any x in G and any f in G ∗ ; or simply φ ( x ) = φ ( x ) + φ ( x ) ; φ ( f ) = φ ( f ) + φ ( f ) . (2.13)In order to find the φ ij ’s and hence all the derivations of D , we are now going to usethe cocycle condition (2.6).For x, y in G ⊂ D we have: φ ([ x, y ]) = φ ([ x, y ]) + φ ([ x, y ]) (2.14) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G and [ φ ( x ) , y ] + [ x, φ ( y )] = [ φ ( x ) + φ ( x ) , y ] + [ x, φ ( y ) + φ ( y )]= [ φ ( x ) , y ] − ad ∗ y ( φ ( x )) + [ x, φ ( y )] + ad ∗ x ( φ ( y )) . (2.15)Comparing (2.14) and (2.15), we first get φ ([ x, y ]) = [ φ ( x ) , y ] + [ x, φ ( y )] , (2.16)for every x, y in G . This means that φ is a derivation of the Lie algebra G .Secondly, for all elements x , y of G , we have φ ([ x, y ]) = ad ∗ x ( φ ( y )) − ad ∗ y ( φ ( x )) . (2.17)Equation (2.17) means that φ : G → G ∗ is a -cocycle of G with values on G ∗ for thecoadjoint action of G on G ∗ .Now we are going to examine the following case: for all x in G and all f in G ∗ , φ ([ x, f ]) = φ ( ad ∗ x f ) , = φ ( ad ∗ x f ) + φ ( ad ∗ x f ) , (2.18)and [ φ ( x ) , f ] + [ x, φ ( f )] = [ φ ( x ) + φ ( x ) , f ] + [ x, φ ( f ) + φ ( f )] , = ad ∗ φ ( x ) f + [ x, φ ( f )] + ad ∗ x (cid:0) φ ( f ) (cid:1) . (2.19)Identifying (2.18) and (2.19) we obtain on the one hand φ ( ad ∗ x f ) = [ x, φ ( f )] , = ad x ( φ ( f )) , for every x in G , and every f in G ∗ . We write the above as φ ◦ ad ∗ x = ad x ◦ φ , for all x in G . That is, φ : G ∗ → G is equivariant (commutes) with respect to the adjointand the coadjoint actions of G on G and G ∗ respectively .We have on the other hand φ ( ad ∗ x f ) = ad ∗ x ( φ ( f )) + ad ∗ φ ( x ) f, (2.20)for all x in G and all f in G ∗ . Formula (2.20) can be rewritten as φ ◦ ad ∗ x − ad ∗ x ◦ φ = ad ∗ φ ( x ) , i.e. for any element x of G , [ φ , ad ∗ x ] = ad ∗ φ ( x ) . Last, for f and g in G ∗ , we have φ ([ f, g ]) = 0 , (2.21) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G and [ φ ( f ) , g ] + [ f, φ ( g )] = [ φ ( f ) + φ ( f ) , g ] + [ f, φ ( g ) + φ ( g )] , = ad ∗ φ ( f ) g − ad ∗ φ ( g ) f. (2.22)From (2.21) and (2.22) it comes that for all elements f, g of G ∗ , ad ∗ φ ( f ) g = ad ∗ φ ( g ) f. Noting α := φ , β := φ , ψ := φ and ξ := φ , we get a proof of Theorem 2.3.1. Remark 2.3.1. ( Notations ) From now on, if G is a Lie algebra, then1. E will stand for the space of linear maps ξ : G ∗ → G ∗ satisfying Equation (2.8), forsome derivation α of G ;2. set G := n φ : D → D , φ ( x, f ) = ( α ( x ) , ξ ( f )) : α ∈ der ( G ) , ξ ∈ E , [ ξ, ad ∗ x ] = ad ∗ α ( x ) , ∀ x ∈ G o ; 3. we may let Q stand for the space of 1-cocycles β : G → G ∗ as in (2.17), whereas Ψ may be used for the space of equivariant linear ψ : G ∗ → G as in (2.9), which satisfy(2.10);4. we will denote by G , the direct sum G := Q ⊕ Ψ of the vector spaces Q and Ψ . Remark 2.3.2. The spaces der ( G ) of derivations of G , Q and Ψ , as in Remark 2.3.1, areall subsets of der ( D ) , as follows. A derivation α of G , an equivariant map ψ in Ψ , and a -cocycle β in Q are respectively seen as the elements φ α , φ ψ , φ β of der ( D ) , with φ α ( x, f ) := ( α ( x ) , − f ◦ α ); φ ψ ( x, f ) := ( ψ ( f ) , φ β ( x, f ) := (0 , β ( x )) , for all ( x, f ) in D . Corollary 2.3.1. Every derivation of G is the restriction to G of a derivation of D . for the Group of Automorphisms of D Lemma 2.3.1. The space E , as in Remark 2.3.1, is a Lie algebra.Namely, if ξ , ξ in E satisfy [ ξ , ad ∗ x ] = ad ∗ α ( x ) and [ ξ , ad ∗ x ] = ad ∗ α ( x ) , for all x in G and some α , α in der( G ), then their Lie bracket [ ξ , ξ ] is in E and satisfies [[ ξ , ξ ] , ad ∗ x ] = ad ∗ [ α ,α ]( x ) . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G Proof. Using Jacobi identity in the Lie algebra G l ( G ∗ ) of endomorphisms of the vector space G ∗ , we get: for any x in G [[ ξ , ξ ] , ad ∗ x ] = [[ ξ , ad ∗ x ] , ξ ] + [ ξ , [ ξ , ad ∗ x ]]= [ ad ∗ α ( x ) , ξ ] + [ ξ , ad ∗ α ( x ) ]= − ad ∗ α ◦ α ( x ) + ad ∗ α ◦ α ( x ) = ad ∗ [ α ,α ]( x ) . Lemma 2.3.2. The space G , as in Remark 2.3.1, is a Lie subalgebra of der ( D ) .Proof. This is a consequence of Lemma 2.3.1. If φ := ( α , ξ ) , and φ := ( α , ξ ) are in G , then [ φ , φ ] = ([ α , α ] , [ ξ , ξ ]) , as can easily be seen, below. For every ( x, f ) in D , wehave [ φ , φ ]( x, f ) = φ (cid:16) α ( x ) , ξ ( f ) (cid:17) − φ (cid:16) α ( x ) , ξ ( f ) (cid:17) = (cid:16) α ◦ α ( x ) , ξ ◦ ξ ( f ) (cid:17) − (cid:16) α ◦ α ( x ) , ξ ◦ ξ ( f ) (cid:17) = (cid:16) [ α , α ]( x ) , [ ξ , ξ ]( f ) (cid:17) . Lemma 2.3.3. Let β and ψ be in Q and Ψ , respectively. Then [ β, ψ ] = ( − ψ ◦ β, β ◦ ψ ) belongs to G , more precisely β ◦ ψ is in E , ψ ◦ β is in der ( G ) and [ β ◦ ψ, ad ∗ x ] = − ad ∗ ψ ◦ β ( x ) ,for any x in G .Proof. First, β being a 1-cocycle is equivalent to β ◦ ad x ( y ) = ad ∗ x ◦ β ( y ) − ad ∗ y ◦ β ( x ) , (2.23)for all x, y in G . Now for every x in G and every f in G ∗ , we have [ β ◦ ψ, ad ∗ x ]( f ) = β ◦ ψ ◦ ad ∗ x ( f ) − ad ∗ x ◦ β ◦ ψ ( f )= β ◦ ad x ◦ ψ ( f ) − ad ∗ x ◦ β ◦ ψ ( f ) , now take y = ψ ( f ) in (2.23) = ad ∗ x ◦ β ◦ ψ ( f ) − ad ∗ ψ ( f ) β ( x ) − ad ∗ x ◦ β ◦ ψ ( f ) , = − ad ∗ ψ ( f ) β ( x ) , take g = β ( x ) in (2.10) = − ad ∗ ψ ◦ β ( x ) f = ad ∗ α ( x ) f, where α = − ψ ◦ β. Next, the proof that ψ ◦ β is in der( G ), is straightforward. Indeed, for every elements x, y in G , we have ψ ◦ β [ x, y ] = ψ (cid:16) ad ∗ x β ( y ) − ad ∗ y β ( x ) (cid:17) = ad x ◦ ψ ◦ β ( y ) − ad y ◦ ψ ◦ β ( x )= [ x, ψ ◦ β ( y )] + [ ψ ◦ β ( x ) , y ] Hence [ β, ψ ] belongs to G , for every β in Q and every ψ in Ψ . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G Let φ := ( α, ξ ) be in G , β : G → G ∗ and ψ : G ∗ → G be respectively in Q and Ψ . Then both [ φ, β ] and [ φ, ψ ] are elements of G , more precisely [ φ, β ] is in Q and [ φ, ψ ] is in Ψ . Moreover, we have [ Q , Q ] = 0 and [Ψ , Ψ] = 0 . Proof. Let φ = ( α, ξ ) be in G , β : G → G ∗ a 1-cocycle and ψ : G ∗ → G an equivariantlinear map. Using φ β and φ ψ as in Remark 2.3.2, we obtain [ φ, φ β ]( x, y ) = φ (cid:16) , β ( x ) (cid:17) − φ β (cid:16) α ( x ) , ξ ( f ) (cid:17) = (cid:16) , ξ ◦ β ( x ) (cid:17) − (cid:16) , β ◦ α ( x ) (cid:17) = (cid:16) , ( ξ ◦ β − β ◦ α )( x ) (cid:17) . Now, let us show that ˜ β := ξ ◦ β − β ◦ α : G → G ∗ is a -cocycle. Indeed, on the one handwe have ξ ◦ β ([ x, y ]) = ξ (cid:16) ad ∗ x β ( y ) − ad ∗ y β ( x ) (cid:17) = (cid:16) [ ξ, ad ∗ x ] + ad ∗ x ◦ ξ (cid:17)(cid:0) β ( y ) (cid:1) − (cid:16) [ ξ, ad ∗ y ] + ad ∗ y ◦ ξ (cid:17)(cid:0) β ( x ) (cid:1) = ad ∗ α ( x ) β ( y ) + ad ∗ x ( ξ ◦ β ( y )) − ad ∗ α ( y ) β ( x ) − ad ∗ y ( ξ ◦ β ( x ))= ad ∗ x ( ξ ◦ β ( y )) − ad ∗ y ( ξ ◦ β ( x )) + ad ∗ α ( x ) β ( y ) − ad ∗ α ( y ) β ( x ) (2.24)On the other hand, we also have β ◦ α ([ x, y ]) = β (cid:0) [ α ( x ) , y ] (cid:1) + β (cid:0) [ x, α ( y )] (cid:1) = ad ∗ α ( x ) β ( y ) − ad ∗ y ( β ◦ α ( x )) + ad ∗ x ( β ◦ α ( y )) − ad ∗ α ( y ) β ( x ) (2.25)Subtracting (2.25) from (2.24), we see that ˜ β [ x, y ] now reads ˜ β [ x, y ] = ad ∗ x ( ξ ◦ β − β ◦ α )( y ) − ad ∗ y ( ξ ◦ β − β ◦ α )( x )= ad ∗ x ˜ β ( y ) − ad ∗ y ˜ β ( x ) Hence ˜ β is an element of Q . In the same way, we also have [ φ, φ ψ ]( x, y ) = φ (cid:0) ψ ( f ) , (cid:1) − φ ψ (cid:0) α ( x ) , ξ ( f ) (cid:1) = (cid:0) α ◦ ψ ( f ) , (cid:1) − (cid:0) ψ ◦ ξ ( f ) , (cid:1) = (cid:0) ( α ◦ ψ − ψ ◦ ξ )( f ) , (cid:1) The linear map ˜ ψ := α ◦ ψ − ψ ◦ ξ : G ∗ → G is equivariant, i.e. is an element of Ψ . Asabove, this is seen by first computing, for every elements x of G and f of G ∗ , α ◦ ψ ( ad ∗ x f ) = α (cid:0) [ x, ψ ( f )] (cid:1) = [ α ( x ) , ψ ( f )] + [ x, α ◦ ψ ( f )] (2.26)and ψ ◦ ξ ( ad ∗ x f ) = ψ ◦ (cid:0) [ ξ, ad ∗ x ] + ad ∗ x ◦ ξ (cid:1) ( f )= ψ (cid:0) ad ∗ α ( x ) f (cid:1) + ψ (cid:0) ad ∗ x ξ ( f ) (cid:1) = (cid:2) α ( x ) , ψ ( f ) (cid:3) + (cid:2) x, ψ ◦ ξ ( f ) (cid:3) , (2.27) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G then subtracting (2.26) and (2.27).Now we have [ φ β , φ β ′ ]( x, f ) = φ β (0 , β ′ ( x )) − φ β ′ (0 , β ( x )) = 0 and [ φ ψ , φ ψ ′ ]( x, f ) = φ ψ ( ψ ′ ( f ) , − φ ψ ′ ( ψ ( f ) , 0) = 0 , for all ( x, f ) in D . In other words, [ Q , Q ] = 0 and [Ψ , Ψ] = 0 . We summarize all the above in the Theorem 2.3.2. Let G be a Lie group and G its Lie algebra. The group Aut( D ) of auto-morphisms of the Lie algebra D of the cotangent bundle T ∗ G of G , is a super symmetric Liegroup. More precisely, its Lie algebra der ( D ) is a Z / Z -graded symmetric (supersymmetric)Lie algebra which decomposes into a direct sum of vector spaces der ( D ) := G ⊕ G , with [ G i , G j ] ⊂ G i + j , i, j ∈ Z / Z = { , } (2.28) where G is the Lie algebra of linear maps φ : D → D , φ ( x, f ) = ( α ( x ) , ξ ( f )) with α aderivation of G and the linear map ξ : G ∗ → G ∗ satisfies, [ ξ, ad ∗ x ] = ad ∗ α ( x ) , (2.29) for any x of G ; and G is the direct sum (as a vector space) of the space Q of -cocycles G → G ∗ and the space Ψ of linear maps G ∗ → G which are equivariant with respect to thecoadjoint and the adjoint representations and satisfy (2.10).Moreover, G ⊕ ˜ G and G ⊕ ˜ G ′ are subalgebras of der( D ) which are Lie superalgebras,i.e. they are Z / Z -graded Lie algebras with the Lie bracket satisfying [ x, y ] = − ( − deg ( x ) deg ( y ) [ y, x ] , where ˜ G := Q and ˜ G ′ := Ψ are Abelian subalgebras of der ( D ) and deg ( x ) = i , if x ∈ G i . Remark 2.3.3. (a) In Propositions 2.3.1 and 2.3.2, we will prove that every element ξ of E is the transpose ξ = ( j − α ) t of the sum of an adjoint-invariant endomorphism j ∈ J and a derivation − α of G . (b) The Lie superalgebras G ⊕ ˜ G and G ⊕ ˜ G ′ respectively correspond to the subalgebrasof all elements of der( D ) which preserve the subalgebra G and the ideal G ∗ of D . TheLie superalgebra G ⊕ ˜ G ′ can be seen as part of the more general case of derivationsof a semi-direct product Lie algebra G ⋉ N which preserve the ideal N and which arediscussed in [65], among other interesting results therein. Let us now have a closer look at maps ξ, ψ and β . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G ξ and Bi-invariant Tensors of Type (1,1) Adjoint-invariant Endomorphisms Linear operators acting on vector fields of a given Lie group G can be seen as fields ofendomorphisms of its tangent spaces. Bi-invariant ones correspond to endomorphisms j : G → G of the Lie algebra G of G , satisfying the condition j [ x, y ] = [ jx, y ] , for all x, y in G .If we denote by ∇ the connection on G given on left invariant vector fields by ∇ x y := 12 [ x, y ] , then using the covariant derivative, we have ∇ j = 0 , (see e.g. [70]). As above, let J := { j : G → G , linear and j [ x, y ] = [ jx, y ] , ∀ x, y ∈ G } . Endowed with the bracket [ j, j ′ ] := j ◦ j ′ − j ′ ◦ j, the space J is a Lie algebra, and indeed a subalgebra of the Lie algebra G l ( G ) of all endo-morphisms of G . In the case where the dimension of G is even and if in addition j satisfies j = − identity ,then ( G, j ) is a complex Lie group. Maps ξ : G ∗ → G ∗ Proposition 2.3.1. Let G be a Lie algebra and α a derivation of G . A linear map ξ ′ : G → G satisfies [ ξ ′ , ad x ] = ad α ( x ) , for every element x of G , if and only if there exists a linear map j : G → G satisfying j ([ x, y ]) = [ j ( x ) , y ] = [ x, j ( y )] , (2.30) for all x, y in G , such that ξ ′ = j + α .Proof. Let α be a derivation and ξ ′ an endomorphism of G satisfying the hypothesis ofProposition 2.3.1, that is, [ ξ ′ , ad x ] = ad α ( x ) = [ α, ad x ] , for any x in G . We then have, [ ξ ′ − α, ad x ] = 0 , (2.31)for any x of G . So the endomorphism j := ξ ′ − α commutes with all adjoint operators.Now a linear map j : G → G commuting with all adjoint operators, satisfies j, ad x ]( y ) = j ([ x, y ]) − [ x, j ( y )] , (2.32)for all elements x, y of G . We also have, j, ad y ]( x ) = j ([ y, x ]) − [ y, j ( x )] , (2.33)for all x, y in G . From (2.32) and (2.33), we have j ([ x, y ]) = [ j ( x ) , y ] = [ x, j ( y )] , for any x, y in G .Thus, (2.31) is equivalent to ξ ′ = j + α , where j satisfies (2.30). On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G The above means that the space J is the centralizer of the space ad G ofinner derivations of G in G l ( G ) := { l : G → G linear } , i.e. J = Z G l ( G ) ( ad G ) := { j : G → G linear and [ j, ad x ] = 0 , ∀ x ∈ G } . Proposition 2.3.2. Let G be a nonabelian Lie algebra and S the space of endomorphisms ξ ′ : G → G such that there exists a derivation α of G and [ ξ ′ , ad x ] = ad α ( x ) for all x ∈ G .Then S is a Lie algebra containing J and der ( G ) as subalgebras. In the case where G has atrivial centre, then S is the semi-direct product S = der ( G ) ⋉ J of J and der ( G ) .The following are equivalent(a) The linear map ξ : G ∗ → G ∗ is an element of E with α as the corresponding derivationof G , i.e. ξ satisfies (2.8) for the derivation α .(b) The transpose ξ t of ξ is of the form ξ t = j − α, where j is in J and α in der( G ).(c) ξ t is an element of S , with corresponding derivation − α. The transposition ξ ξ t of linear maps is an anti-isomorphism between the Lie algebras E and S . Proof. Using the same argument as in Lemma 2.3.1, if [ ξ ′ , ad x ] = ad α ( x ) and [ ξ ′ , ad x ] = ad α ( x ) , for every x in G , then [[ ξ ′ , ξ ′ ] , ad x ] = ad [ α ,α ]( x ) , for any element x of G . Thus S isa Lie algebra. From Proposition 2.3.1, there exist j i in J such that ξ ′ i = α i + j i , i = 1 , . Obviously, S contains J and der ( G ) . Thus, as a vector space, S decomposes as S = der ( G )+ J . Now, the Lie bracket in S reads [ ξ ′ , ξ ′ ] = [ α + j , α + j ] = [ α , α ] + [ α , j ] + [ j , α ] + [ j , j ] (2.34)Of course, [ α , α ] is in der ( G ) . From Section 2.3.3, we know that J is a Lie algebra, hence [ j , j ] is in J . It is easy to check that [ α, j ] ∈ J , (2.35)for all α in der ( G ) and for all j in J . Indeed, the following holds [ α, j ]([ x, y ]) = α (cid:0) [ j ( x ) , y ] (cid:1) − j (cid:0) [ α ( x ) , y ] + [ x, α ( y )] (cid:1) = [ α ◦ j ( x ) , y ] + [ j ( x ) , α ( y )] − [ j ◦ α ( x ) , y ] − [ j ( x ) , α ( y )]= [[ α, j ]( x ) , y ] , (2.36)for all x, y in G . The intersection der( G ) ∩ J is made of elements j of J whose image Im( j )is a subset of the centre Z ( G ) of G . Hence if Z ( G ) = 0 , then S = der( G ) ⊕ J and as a Liealgebra, S = der( G ) ⋉ J . Using this decomposition, we can also rewrite (2.34) as [ ξ ′ , ξ ′ ] = [( α , j ) , ( α , j )] = ([ α , α ] , [ j , j ] + [ α , j ] + [ j , α ]) (2.37)The equivalence between (b) and (c) comes directly from Proposition 2.3.1.Now let ξ ∈ E , with [ ξ, ad ∗ x ] = ad ∗ α ( x ) , α ∈ der ( G ) , then − ad α ( x ) = [ ξ, ad ∗ x ] t = − [ ξ t , ( ad ∗ x ) t ] = [ ξ t , ad x ] On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G Hence ξ t ∈ S , with ad α ′ ( x ) = [ ξ t , ad x ] , for all x ∈ G , where α ′ := − α . Thus, (a) implies (c).From Proposition 2.3.1, there exist j ∈ J such that ξ t = − α + j. Now it is straightforwardthat if (b) ξ t = − α + j with α a derivation and j in J , then ξ satisfies [ ξ, ad ∗ x ] = ad ∗ α ( x ) , forall x in G . Hence (c) implies (a).Of course, we also know that [ ξ , ξ ] t = − [ ξ t , ξ t ] , for every ξ , ξ ∈ E . Lemma 2.3.5. Let ξ ′ : G → G be a linear map such that there exists α : G → G linear and [ ξ ′ , ad x ] = ad α ( x ) , for all x in G . Then ξ ′ preserves every ideal A of G satisfying [ A , A ] = A . In particular, if G is semi-simple and G = s ⊕ s ⊕ . . . ⊕ s p is a decomposition of G into asum of simple ideals s , . . . , s p , then ξ ′ ( s i ) ⊂ s i , for i = 1 , . . . , p. Proof. The proof is straightforward. Indeed, every element x of an ideal A satisfying thehypothesis of Lemma 2.3.5, is a finite sum of the form x = X i [ x i , y i ] where x i , y i are allelements of A . But as A is an ideal, ξ ′ ([ x i , y i ]) = ξ ′ ◦ ad x i ( y i ) = (cid:0) [ ξ ′ , ad x i ] + ad x i ◦ ξ ′ (cid:1) ( y i )= (cid:0) ad α ( x i ) + ad x i ◦ ξ ′ (cid:1) ( y i ) = [ α ( x i ) , y i ] + [ x i , ξ ′ ( y i )] is again an element of A . Hence we have ξ ′ ( x ) = X i (cid:0) [ α ( x i ) , y i ] + [ x i , ξ ′ ( y i )] (cid:1) is in A . ψ : G ∗ → G Let G be a Lie algebra. In this section, we would like to explore properties of the space Ψ of linear maps ψ : G ∗ → G which are equivariant with respect to the adjoint and thecoadjoint actions of G on G and G ∗ respectively and satisfy: for all f, g in G ∗ , ad ∗ ψ ( f ) g = ad ∗ ψ ( g ) f. Lemma 2.3.6. Let G be a Lie algebra and ψ an element of Ψ . Then,(a) Imψ is an Abelian ideal of G and we have ψ ( ad ∗ ψ ( g ) f ) = 0 , for every f, g in G ∗ ;(b) ψ sends closed forms on G in the center of G ;(c) [ Imψ, G ] ⊂ ker f , for all f in ker ψ ;(d) the map ψ cannot be invertible if G is not Abelian.Proof. (a) For every elements f of G ∗ and x of G , we have, [ ψ ( f ) , x ] = − ( ad x ◦ ψ )( f ) = − ( ψ ◦ ad ∗ x )( f ) ∈ Imψ. Hence Im ( ψ ) is an ideal of G .Now, for every f, g in G ∗ , since ψ ( f ) and ψ ( g ) are elements of G , we also have ψ ◦ ad ∗ ψ ( f ) = ad ψ ( f ) ◦ ψ and ψ ◦ ad ∗ ψ ( g ) = ad ψ ( g ) ◦ ψ . On the one hand, (cid:0) ψ ◦ ad ∗ ψ ( f ) (cid:1) ( g ) = (cid:0) ad ψ ( f ) ◦ ψ (cid:1) ( g ) ψ ( ad ∗ ψ ( f ) g ) = [ ψ ( f ) , ψ ( g )] (2.38) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G On the other hand, (cid:0) ψ ◦ ad ∗ ψ ( g ) (cid:1) ( f ) = (cid:0) ad ψ ( g ) ◦ ψ (cid:1) ( f ) ψ ( ad ∗ ψ ( g ) f ) = [ ψ ( g ) , ψ ( f )] (2.39)Using (2.38) and (2.39) we get [ ψ ( f ) , ψ ( g )] = ψ ( ad ∗ ψ ( f ) g ) = ψ ( ad ∗ ψ ( g ) f ) = [ ψ ( g ) , ψ ( f )] (2.40)Equation (2.40) implies the following [ ψ ( f ) , ψ ( g )] = ψ ( ad ∗ ψ ( g ) f ) = 0 , (2.41)for all elements f, g of G ∗ . So we have proved (a).(b) Let f be a closed form on G , that is, f in G ∗ and ad ∗ x f = 0 , for all x in G . The relation(2.10) implies that ad ∗ ψ ( f ) g = 0 , for any g in G ∗ . Thus, for any element y of G and anyelement g of G ∗ , g ([ ψ ( f ) , y ]) = 0 , and hence [ ψ ( f ) , y ] = 0 , for all y in G . In other words ψ ( f ) belongs to the center of G .(c) If f ∈ ker ψ , then ad ∗ ψ ( g ) f = ad ∗ ψ ( f ) g = 0 , for any g in G ∗ , or equivalently, for every x in G and g in G ∗ , f ([ ψ ( g ) , x ]) = 0 . It follows that [ Imψ, G ] ⊂ ker f , for every f of ker ψ .(d) From (a), the map ψ satisfies ψ ( ad ∗ ψ ( g ) f ) = 0 , for any f, g in G ∗ . There are two possi-bilities here:(i) either there exist f, g in G ∗ such that ad ∗ ψ ( g ) f = 0 , in which case ad ∗ ψ ( g ) f belongs to ker ψ = 0 and thus ψ is not invertible;(ii) or else, suppose ad ∗ ψ ( g ) f = 0 , for all f, g in G ∗ . This implies that ψ ( g ) belongs to thecenter of G for every g in G ∗ . In other words, the center of G contains Im ( ψ ) . Butsince G is not Abelian, the center of G is different from G , hence ψ is not invertible. Lemma 2.3.7. The space of equivariant maps ψ : G ∗ → G bijectively corresponds to thatof G -invariant bilinear forms on the G -module G ∗ for the coadjoint representation.Proof. Indeed, each such ψ defines a unique coadjoint-invariant bilinear form h , i ψ on G ∗ as follows: h f, g i ψ := h ψ ( f ) , g i , (2.42)for all f, g in G ∗ , where the right hand side is the duality pairing h f, x i = f ( x ) , x in G , f in G ∗ , as above. The coadjoint-invariance reads h ad ∗ x f, g i ψ + h f, ad ∗ x g i ψ = 0 , (2.43)for all x in G and all f, g in G ∗ ; and is due to the simple equalities h ad ∗ x f, g i ψ = h ψ ( ad ∗ x f ) , g i = h ad x ψ ( f ) , g i = −h ψ ( f ) , ad ∗ x g i = −h f, ad ∗ x g i ψ . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G Conversely, every G -invariant bilinear form h , i on G ∗ gives rise to a unique linear map ψ : G ∗ → G which is equivariant with respect to the adjoint and coadjoint representationsof G , by the formula h ψ ( f ) , g i := h f, g i . (2.44)If ψ is symmetric or skew-symmetric, then so is h , i ψ and vice versa. Otherwise, h , i ψ canbe decomposed into a symmetric and a skew-symmetric parts h , i ψ,s and h , i ψ,a respectively,defined by the following formulas: h f, g i ψ,s := 12 h h f, g i ψ + h g, f i ψ i , (2.45) h f, g i ψ,a := 12 h h f, g i ψ − h g, f i ψ i . (2.46)The symmetric and skew-symmetric parts h , i ,s and h , i ,a of a G -invariant bilinear form h , i , are also G -invariant. From a remark in p. 2297 of [60], the radical Rad h , i := { f ∈ G ∗ , h f, g i = 0 , ∀ g ∈ G ∗ } of a G -invariant form h , i , contains the coadjoint orbits of all itspoints. G → G ∗ . The 1-cocycles for the coadjoint representation of a Lie algebra G are linear maps β : G → G ∗ satisfying the cocycle condition β ([ x, y ]) = ad ∗ x β ( y ) − ad ∗ y β ( x ) , for everyelements x, y of G . To any given 1-cocycle β , corresponds a bilinear form Ω β on G , by the formula Ω β ( x, y ) := h β ( x ) , y i , (2.47)for all x, y in G , where h , i is again the duality pairing between elements of G and G ∗ .The bilinear form Ω β is skew-symmetric (resp. symmetric, nondegenerate) if and onlyif β is skew-symmetric (resp. symmetric, invertible).Skew-symmetric such cocycles β are in bijective correspondence with closed 2-formsin G , via the formula (2.47). In this sense, the cohomology space H ( D , D ) contains thesecond cohomology space H ( G , R ) of G with coefficients in R for the trivial action of G on R . Hence, H ( D , D ) somehow contains the second space H inv ( G, R ) of left invariant deRham cohomology H ∗ inv ( G, R ) of any Lie group G with Lie algebra G . Invertible skew-symmetric ones, when they exist, are those giving rise to symplecticforms or equivalently to invertible solutions of the Classical Yang-Baxter Equation. Thestudy and classification of the solutions of the Classical Yang-Baxter Equation is a stillopen problem in Geometry, Theory of integrable systems. In Geometry, they give rise tovery interesting structures in the framework of Symplectic Geometry, Affine Geometry,Theory of Homogeneous Kähler domains, (see e.g. [28] and references therein).If G is semi-simple, then every cocycle β is a coboundary, that is, there exists f β in G ∗ such that β ( x ) = − ad ∗ x f β , for any x in G . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 roup of Automorphisms of D := T ∗ G H ( D , D ) Following Remarks 2.3.1 and 2.3.2, we can embed der ( G ) as a subalgebra der ( G ) of der ( D ) ,using the linear map α φ α , with φ α ( x, f ) = ( α ( x ) , f ◦ α ) . In the same way, we haveconstructed Q and Ψ as subspaces of der ( D ) . Likewise, J t := { j t , where j ∈ J } is seen asa subspace of der ( D ) , via the linear map j t φ j , with φ j ( x, f ) = (0 , f ◦ j ) . We give the following summary. Theorem 2.3.3. The first cohomology space H ( D , D ) of the (Chevalley-Eilenberg) coho-mology associated with the adjoint action of D on itself, satisfies H ( D , D ) ∼ = H ( G , G ) ⊕ J t ⊕ H ( G , G ∗ ) ⊕ Ψ , where H ( G , G ) and H ( G , G ∗ ) are the first cohomology spaces associated with the adjointand coadjoint actions of G , respectively; and J t := { j t , j ∈ J } (space of transposes ofelements of J ).If G is semi-simple, then Ψ = { } and thus H ( D , D ) ∼ = J t . Moreover, we have J ∼ = R p , where p is the number of the simple ideals s i of G such that G = s ⊕ . . . ⊕ s p . Hence, of course, H ( D , D ) ∼ = R p . If G is a compact Lie algebra, with centre Z ( G ) , we get H ( G , G ) ∼ = End ( Z ( G )) , J ∼ = R p ⊕ End ( Z ( G )) ,H ( G , G ∗ ) ∼ = L ( Z ( G ) , Z ( G ) ∗ ) , Ψ ∼ = L ( Z ( G ) ∗ , Z ( G )) . Hence, we get H ( D , D ) ∼ = ( End ( R k )) ⊕ R p , where k is the dimension of the centre of G , and p is the number of the simple componentsof the derived ideal [ G , G ] of G . Here, if E, F are vector spaces, L ( E, F ) is the space oflinear maps E → F. The proof of Theorem 2.3.3 is given by Proposition 2.3.3 below and different lemmasand propositions, discussed in Section 2.4.For the purpose of this investigation, we have favored a direct approach to exhibitdetailed calculations of the first cohomology space, instead of the traditional powerfulspectral sequences method commonly applied in the more general setting of locally trivialfiber bundles (see e.g. [44], [65]). Some parts of Theorem C can also be seen as a refinement,using our direct approach, of some already known results ([44]).As a vector space, der ( D ) is isomorphic to the direct sum der ( G ) ⊕ J t ⊕ Q ⊕ Ψ by Φ : der ( G ) ⊕ J t ⊕ Q ⊕ Ψ → der ( D ); ( α, j t , β, ψ ) φ α + φ j + φ β + φ ψ . (2.48) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras In this isomorphism, we have Φ( der ( G ) ⊕ J t ) = der ( G ) ⊕ J t = G and Φ( Q ⊕ Ψ) = G .Now an exact derivation of D , i.e. a -coboundary for the Chevalley-Eilenberg coho-mology associated with the adjoint action of D on D , is of the form φ = ∂v = ad v forsome element v := ( x , f ) of the D -module D . That is, φ ( x, f ) = ( α ( x ) , β ( x ) + ξ ( f )) , where α ( x ) := [ x , x ] , β ( x ) = − ad ∗ x f , ξ ( f ) = ad ∗ x f. As we can see φ = φ α + φ β = Φ( α , , β , and Proposition 2.3.3. The linear map Φ in (2.48) induces an isomorphism ¯Φ in cohomology,between the spaces H ( G , G ) ⊕ J t ⊕ H ( G , G ∗ ) ⊕ Ψ and H ( D , D ) . Proof. The isomorphism in cohomology simply reads ¯Φ( class ( α ) , j t , class ( β ) , ψ ) = class ( φ α + φ j + φ β + φ ψ ) . In this section, we prove that if a Lie algebra G is orthogonal, then the Lie algebra der ( G ) ofits derivations and the Lie algebra J of linear maps j : G → G satisfying j [ x, y ] = [ jx, y ] , forevery x, y in G , completely characterize the Lie algebra der ( D ) of derivations of D := G ⋉ G ∗ ,and hence the group of automorphisms of the cotangent bundle of any connected Lie groupwith Lie algebra G . We also show that J is isomorphic to the space of adjoint-invariantbilinear forms on G . Let ( G , µ ) be an orthogonal Lie algebra and consider the isomorphism θ : G → G ∗ of G -modules, given by h θ ( x ) , y i := µ ( x, y ) , as in Section 2.2.1.Of course, θ − is an equivariant map. But if G is not Abelian, invertible equivariant linearmaps do not contribute to the space of derivations of D , as discussed in Lemma 2.3.6.We pull coadjoint-invariant bilinear forms B ′ on G ∗ back to adjoint-invariant bilinearforms on G , as follows B ( x, y ) := B ′ ( θ ( x ) , θ ( y )) . Indeed, we have B ([ x, y ] , z ) = B ′ (cid:0) θ ([ x, y ]) , θ ( z ) (cid:1) = B ′ (cid:0) ad ∗ x θ ( y ) , θ ( z ) (cid:1) = − B ′ (cid:0) θ ( y ) , ad ∗ x θ ( z ) (cid:1) = − B ( y, [ x, z ]) . Proposition 2.4.1. If a Lie algebra G is orthogonal, then there is an isomorphism betweenany two of the following vector spaces:(a) the space J of linear maps j : G → G satisfying j [ x, y ] = [ jx, y ] , for every x, y in G ; On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras (b) the space of linear maps ψ : G ∗ → G which are equivariant with respect to the coadjointand the adjoint representations of G ;(c) the space of bilinear forms B on G which are adjoint-invariant, i.e. B ([ x, y ] , z ) + B ( y, [ x, z ]) = 0 , (2.49) for all x, y, z in G ;(d) the space of bilinear forms B ′ on G ∗ which are coadjoint-invariant, i.e. B ′ ( ad ∗ x f, g ) + B ′ ( f, ad ∗ x g ) = 0 , (2.50) for all x in G , f, g in G ∗ .Proof. • The linear map ψ ψ ◦ θ is an isomorphism between the space of equivariantlinear maps ψ : G ∗ → G and the space J . Indeed, if ψ is equivariant, we have ψ ◦ θ ([ x, y ]) = − ψ ( ad ∗ y θ ( x )) = − ad y ψ ( θ ( x )) = [ ψ ◦ θ ( x ) , y ] . Hence ψ ◦ θ is in J . Conversely, if j is in J , then j ◦ θ − is equivariant, as it satisfies j ◦ θ − ◦ ad ∗ x = j ◦ ad x ◦ θ − = ad x ◦ j ◦ θ − . This correspondence is obviously linear and invertible. Hence we get the isomorphismbetween (a) and (b). • The isomorphism between the space J of adjoint-invariant endomorphisms and adjoint-invariant bilinear forms is given as follows j ∈ J B j , where B j ( x, y ) := µ ( j ( x ) , y ) . (2.51)for any x, y in G . We have, for any x, y, z in G B j ([ x, y ] , z ) := µ ( j ([ x, y ]) , z ) = µ ([ x, j ( y )] , z ) = − µ ( j ( y ) , [ x, z ]) = − B j ( y, [ x, z ]) . Conversely, if B is an adjoint-invariant bilinear form on G , then the endomorphism j ,defined by µ ( j ( x ) , y ) := B ( x, y ) (2.52)is an element of J , as it satisfies µ ( j ([ x, y ]) , z ) := B ([ x, y ] , z ) = B ( x, [ y, z ]) = µ ( j ( x ) , [ y, z ]) = µ ([ j ( x ) , y ] , z ) , for all elements x, y, z of G . • From Lemma 2.3.7, the space of equivariant linear maps ψ bijectively corresponds tothat of coadjoint-invariant bilinear forms on G ∗ , via ψ 7→ h , i ψ . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras Now, suppose ψ is skew-symmetric. Let ω ψ denote the corresponding skew-symmetricbilinear form on G : ω ψ ( x, y ) := µ ( ψ ◦ θ ( x ) , y ) , (2.53)for all x, y in G . Then, ω ψ is adjoint-invariant. If we denote by ∂ the Chevalley-Eilenbergcoboundary operator, that is, ( ∂ω ψ )( x, y, z ) = − (cid:0) ω ψ ([ x, y ] , z ) + ω ψ ([ y, z ] , x ) + ω ψ ([ z, x ] , y ) (cid:1) , the following formula holds true ( ∂ω ψ )( x, y, z ) = − ω ψ ([ x, y ] , z ) . for all x, y, z in G . Corollary 2.4.1. The following are equivalent.(a) ω ψ is closed;(b) ψ ◦ θ ([ x, y ]) = 0 , for all x, y in G ;(c) Im ( ψ ) is in the centre of G .In particular, if dim[ G , G ] ≥ dim G − , then ω ψ is closed if and only if ψ = 0 . Proof. The above equality also reads ∂ω ψ ( x, y, z ) = − ω ψ ([ x, y ] , z ) = − µ ( ψ ◦ θ ([ x, y ]) , z ) , (2.54)for all x, y, z in G ; and gives the proof that (a) and (b) are equivalent. In particular, if G = [ G , G ] then, obviously ∂ω ψ = 0 if and only if ψ = 0 , as θ is invertible.Now suppose dim[ G , G ] = dim G − and set G = R x ⊕ [ G , G ] , for some element x of G . If ω ψ is closed, we already know that ψ ◦ θ vanishes on [ G , G ] . Below, we show that, it alsodoes on R x . Indeed, the formula − ω ψ ([ x, y ] , x ) = ω ψ ( x , [ x, y ]) = µ ( ψ ◦ θ ( x ) , [ x, y ]) , (2.55)for all x, y in G , obtained by taking z = x in (2.54), coupled with the obvious equality ω ψ ( x , x ) = µ ( ψ ◦ θ ( x ) , x ) , are equivalent to ψ ◦ θ ( x ) satisfying µ ( ψ ◦ θ ( x ) , x ) = 0 for all x in G . As µ is nondegenerate, this means that ψ ◦ θ ( x ) = 0 . Hence ψ ◦ θ = 0 , orequivalently ψ = 0 . Now, as every f in G ∗ is of the form f = θ ( y ) , for some y in G , the formula ψ ◦ θ ([ x, y ]) = ψ ◦ ad ∗ x θ ( y ) = ad x ◦ ψ ◦ θ ( y ) = [ x, ψ ◦ θ ( y )] , ∀ x, y ∈ G . (2.56)shows that ψ ◦ θ ([ x, y ]) = 0 , for all x, y on G if and only if Im ( ψ ) is a subset of the centerof G . Thus, (b) is equivalent to (c).Now we pull every element ξ of E back to an endomorphism ξ ′ of G given by the formula ξ ′ := θ − ◦ ξ ◦ θ. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras Let ( G , µ ) be an orthogonal Lie algebra and G ∗ its dual space. Define θ : G → G ∗ by h θ ( x ) , y i := µ ( x, y ) , as in Relation (1.2), and let E and S stand for the sameLie algebras as above. The linear map Q : ξ ξ ′ := θ − ◦ ξ ◦ θ is an isomorphism of Liealgebras between E and S . Proof. Let ξ be in E , with [ ξ, ad ∗ x ] = ad ∗ α ( x ) , for every x in G . The image Q ( ξ ) =: ξ ′ of ξ, satisfies, for any x in G , [ ξ ′ , ad x ] := ξ ′ ◦ ad x − ad x ◦ ξ ′ := θ − ◦ ξ ◦ θ ◦ ad x − ad x ◦ θ − ◦ ξ ◦ θ = θ − ◦ ξ ◦ ad ∗ x ◦ θ − θ − ◦ ad ∗ x ◦ ξ ◦ θ = θ − ◦ ( ξ ◦ ad ∗ x − ad ∗ x ◦ ξ ) ◦ θ = θ − ◦ ad ∗ α ( x ) ◦ θ, since [ ξ, ad ∗ x ] = ad ∗ α ( x ) . = θ − ◦ θ ◦ ad α ( x ) , using (1 . . = ad α ( x ) . Now we have [ Q ( ξ ) , Q ( ξ )]) = Q ([ ξ , ξ ]) for all ξ , ξ in E , as seen below. [ Q ( ξ ) , Q ( ξ )]) := Q ( ξ ) Q ( ξ ) − Q ( ξ ) Q ( ξ ):= θ − ◦ ξ ◦ θ ◦ θ − ◦ ξ ◦ θ − θ − ◦ ξ ◦ θ ◦ θ − ◦ ξ ◦ θ = θ − ◦ [ ξ , ξ ] ◦ θ = Q ([ ξ , ξ ]) . (2.57) Proposition 2.4.3. The linear map P : β D β := θ − ◦ β, is an isomorphism betweenthe space of cocycles β : G → G ∗ and the space der ( G ) of derivations of G . Proof. The proof is straightforward. If β : G → G ∗ is a cocycle, then the linear map D β : G → G , x θ − ( β ( x )) is a derivation of G , as we have D β [ x, y ] = θ − (cid:0) ad ∗ x β ( y ) − ad ∗ y β ( x ) (cid:1) = [ x, θ − ( β ( y ))] − [ y, θ − ( β ( x ))] . Conversely, if D is a derivation of G , then the linear map β D := P − ( D ) = θ ◦ D : G → G ∗ , is 1-cocycle. Indeed we have: for every x, y in G β D [ x, y ] = θ ([ Dx, y ] + [ x, Dy ]) = − ad ∗ y ( θ ◦ D ( x )) + ad ∗ x ( θ ◦ D ( y )) . Suppose now G is semi-simple, then every derivation is inner. Thus in particular, the deriva-tion φ obtained in (2.16), is of the form φ = ad x , (2.58) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras for some x in G . The semi-simplicity of G also implies that the -cocycle φ obtained in(2.17) is a coboundary. That is, there exists an element f of G ∗ such that φ ( x ) = − ad ∗ x f , (2.59)for all x in G . Here is a direct corollary of Lemma 2.3.6. Proposition 2.4.4. If G is a semi-simple Lie algebra, then every linear map ψ : G ∗ → G which is equivariant with respect to the adjoint and coadjoint actions of G and satisfies(2.10), is necessarily identically equal to zero.Proof. A Lie algebra is semi-simple if and only if it contains no nonzero proper Abelianideal. But from Lemma 2.3.6, Im ( ψ ) must be an Abelian ideal of G . So Im ( ψ ) = { } andhence ψ = 0 . Remark 2.4.1. From Propositions 2.3.3 and 2.4.4, the cohomology space H ( D , D ) iscompletely determine by the space J of endomorphisms j with j ([ x, y ]) = [ j ( x ) , y ] , for all x, y in G , or equivalently, by the space of adjoint-invariant bilinear forms on G . Corollary 2.4.2. If G is a semi-simple Lie group with Lie algebra G , then the space ofbi-invariant bilinear forms on G is of dimension dimH ( D , D ) . Proposition 2.4.5. Suppose G is a simple Lie algebra. Then,(a) every linear map j : G → G in J , is of the form j ( x ) = λx , for some λ in R ;(b) every element ξ of E is of the form ξ = ad ∗ x + λId G ∗ , (2.60) for some x in G and λ in R .Proof. The part (a) is obtained from relation (2.31) and the Schur’s lemma.From Propositions 2.3.1 and 2.3.2, for every ξ in E , there exist α in der ( G ) and j in J suchthat ξ t = α + j . As G is simple and from part (a), there exist x in G and λ in R such that ξ t = ad x + λId G . We also have the following. Proposition 2.4.6. Let G be a simple Lie group with Lie algebra G . Let D := G ⋉ G ∗ bethe Lie algebra of the cotangent bundle T ∗ G of G . Then, the first cohomology space of D with coefficients in D is H ( D , D ) ∼ = R .Proof. Indeed, a derivation φ : D → D can be written: for every element ( x, f ) of D , φ ( x, f ) = ([ x , x ] , ad ∗ x f − ad ∗ x f + λf ) , (2.61)where x and f are fixed elements in G and G ∗ respectively. The inner derivations are thosewith λ = 0 . It follows that the first cohomology space of D with values in D is given by H ( D , D ) = { φ : D → D : φ ( x, f ) = (0 , λf ) , λ ∈ R } = { λ (0 , Id G ∗ ) , λ ∈ R } = R Id G ∗ On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras As a direct consequence, we get Corollary 2.4.3. If G is a semi-simple Lie algebra over R , then dim H ( D , D ) = p, where p stands for the number of simple components of G , in its decomposition into a direct sum G = s ⊕ · · · ⊕ s p of simple ideals s , . . . , s p . Consider a semi-simple Lie algebra G and set G = s ⊕ s ⊕ · · · ⊕ s p , p ∈ N ∗ , where s i , i = 1 , . . . , p are simple Lie algebras. From Lemma 2.3.5, ξ ′ preserves each s i . Thusfrom Proposition 2.4.5, the restriction ξ ′ i of ξ ′ to each s i , i = 1 , , . . . , p equals ξ ′ i = ad x i + λ i Id s i , for some x i in s i and a real number λ i . Hence, ξ ′ = ad x ⊕ pi =1 λ i Id s i , where x = x + x + · · · + x p ∈ s ⊕ s ⊕ · · · ⊕ s p and ⊕ pi =1 λ i Id s i acts on s ⊕ s ⊕ · · · ⊕ s p asfollows: ( ⊕ pi =1 λ i Id s i )( x + x + · · · + x p ) = λ x + λ x + · · · + λ p x p . In particular, we have proved Corollary 2.4.4. Consider the decomposition of a semi-simple Lie algebra G into a sum G = s ⊕ s ⊕ · · · ⊕ s p , of simple Lie algebras s i , i = 1 , . . . , p ∈ N ∗ . If a linear map j : G → G satisfies j [ x, y ] = [ jx, y ] , then there exist real numbers λ , . . . λ p such that j = λ id s ⊕ · · · ⊕ λ p id s p . More precisely j ( x + · · · + x p ) = λ x + · · · + λ p x p , if x i is in s i , i = 1 , . . . , p. Now, we already know from Proposition 2.4.4, that each ψ vanishes identically. So a -cocycle of D is given by: φ ( x, f ) = (cid:16) [ x , x ] , ad ∗ x f − ad ∗ x f + p X i =1 λ i f i (cid:17) (2.62)for every x in G and every f := f + f + · · · + f p in s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p = G ∗ , where x is in G , f is in G ∗ and λ i , i = 1 , .., p , are real numbers. We then have, Proposition 2.4.7. Let G be a semi-simple Lie group with Lie algebra G over R . Let D := G ⋉ G ∗ be the cotangent Lie algebra of G . Then, the first cohomology space of D with coefficients in D is given by H ( D , D ) ∼ = R p , where p is the number of the simplecomponents of G . It is well known that a compact Lie algebra G decomposes as the direct sum G = [ G , G ] ⊕ Z ( G ) of its derived ideal [ G , G ] and its centre Z ( G ) , with [ G , G ] semi-simple and compact. Thisyields a decomposition G ∗ = [ G , G ] ∗ ⊕ Z ( G ) ∗ of G ∗ into a direct sum of the dual spaces [ G , G ] ∗ , Z ( G ) ∗ of [ G , G ] and Z ( G ) respectively, where [ G , G ] ∗ (resp. Z ( G ) ∗ ) is identified withthe space of linear forms on G which vanish on Z ( G ) (resp. [ G , G ] ). On the other hand, [ G , G ] also decomposes into as a direct sum [ G , G ] = s ⊕ . . . ⊕ s p of simple ideals s i . From On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras Theorem 2.3.1, a derivation of the Lie algebra D := G ⋉ G ∗ of the cotangent bundle of G has the following form φ ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ξ ( f ) (cid:17) , with conditions listed inTheorem 2.3.1.Let us look at the equivariant maps ψ : G ∗ → G satisfying ad ∗ ψ ( f ) g = ad ∗ ψ ( g ) f , for every f, g in G ∗ . From Lemma 2.3.6, Im ( ψ ) is an Abelian ideal of G ; thus Im ( ψ ) ⊂ Z ( G ) . Asconsequence, we have ψ ( ad ∗ x f ) = [ x, ψ ( f )] = 0 , for every x of G and every f of G ∗ . Lemma 2.4.1. Let (˜ G , µ ) be an orthogonal Lie algebra satisfying ˜ G = [˜ G , ˜ G ] . Then, every g in ˜ G ∗ is a finite sum of elements of the form g i = ad ∗ ¯ x i ¯ g i , for some ¯ x i in ˜ G , ¯ g i in ˜ G ∗ . Proof. Indeed, consider an isomorphism θ : ˜ G → ˜ G ∗ of ˜ G -modules. For every g in ˜ G ∗ , thereexists x g in ˜ G such that g = θ ( x g ) . But as ˜ G = [˜ G , ˜ G ] , we have x g = [ x , y ] + . . . + [ x s , y s ] for some x i , y i in ˜ G . Thus g = θ ([ x , y ]) + . . . + θ ([ x s , y s ])= ad ∗ x θ ( y ) + . . . + ad ∗ x s θ ( y s )= ad ∗ ¯ x ¯ g + . . . + ad ∗ ¯ x s ¯ g s , where ¯ x i = x i and ¯ g i = θ ( y i ) . A semi-simple Lie algebra being orthogonal (with, e.g. its Killing form as µ ), fromLemma 2.4.1 and the equality ψ ( ad ∗ x f ) = 0 , for all x in G , f in G ∗ , each ψ in Ψ vanishes on [ G , G ] ∗ . Of course, the converse is true. Every linear map ψ : G ∗ → G with Im ( ψ ) ⊂ Z ( G ) and ψ ([ G , G ] ∗ ) = 0 , is in Ψ . Hence we can make the following identification. Lemma 2.4.2. Let G be a compact Lie algebra, with centre Z ( G ) . Then Ψ is isomorphicto the space L ( Z ( G ) ∗ , Z ( G )) of linear maps Z ( G ) ∗ → Z ( G ) . The restriction of the cocycle β to the semi-simple ideal [ G , G ] is a coboundary, thatis, there exits an element f in G ∗ such that for every x in [ G , G ] , β ( x ) = − ad ∗ x f . Nowfor x in Z ( G ) , one has β [ x , y ] = − ad ∗ y β ( x ) , for all y of G , since x is in Z ( G ) . Inother words, β ( x )([ y, z ]) = 0 , for all y, z in G . That is, β ( x ) vanishes on [ G , G ] for every x ∈ Z ( G ) . Hence, we write β ( x ) = − ad ∗ x f + η ( x ) , for all x := x + x in [ G , G ] ⊕ Z ( G ) , where η : Z ( G ) → Z ( G ) ∗ is a linear map. This simplymeans the following. Lemma 2.4.3. Let G be a compact Lie algebra, with centre Z ( G ) . Then the first space H ( G , G ∗ ) of the cohomology associated with the coadjoint action of G , is isomorphic to thespace L ( Z ( G ) , Z ( G ) ∗ ) . We have already seen that ξ is such that ξ t = α + j , where α is a derivation of G and j is in J . Both α and j preserve each of [ G , G ] and Z ( G ) . Thus we can write α = ad x ⊕ ϕ, for some x ∈ [ G , G ] , where ϕ is in End ( Z ( G )) . Here α acts on an element x := x + x ,where x is in [ G , G ] , x belongs to Z ( G ) , as follows: α ( x ) = ( ad x ⊕ ϕ )( x + x ):= ad x x + ϕ ( x ) . We summarize this as On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ome Possible Applications and Open Problems If G is a compact Lie algebra with centre Z ( G ) , then H ( G , G ) ∼ = End ( Z ( G )) . (2.63)Now, suppose for the rest of this section, that G is a compact Lie algebra. We write j = ⊕ pi =1 λ i Id s i ⊕ ρ, where ρ : Z ( G ) → Z ( G ) is a linear map and j acts on an element x := x + x as follows: j ( x ) = (cid:16) p M i =1 λ i Id s i ⊕ ρ (cid:17) ( x + x + · · · + x p + x ) = p X i =1 λ i x i + ρ ( x ) where x := x + x + · · · + x p is in [ G , G ] , x is in Z ( G ) and x i belongs to s i . Hence, Lemma 2.4.5. If G is a compact Lie algebra with centre Z ( G ) , then J ∼ = R p ⊕ End ( Z ( G )) , where p is the number of simple components of [ G , G ] . So, the expression of ξ now reads ξ = (cid:2) − ad ∗ x + ( ⊕ pi =1 λ i Id s ∗ i ) (cid:3) ⊕ ϕ ′′ , with ( ϕ ′′ ) t ( x ) = ρ ( x ) + ϕ ( x ) , for all x in Z ( G ) , where x is in [ G , G ] , λ i is in R , for all i = 1 , , . . . , p .By identifying End ( Z ( G )) , L ( Z ( G ) ∗ , Z ( G )) and L ( Z ( G ) , Z ( G ) ∗ ) to End ( R k ) , we get H ( D , D ) = ( End ( R k )) ⊕ R p . (2.64) Given two left or right invariant structures of the same ‘nature’ (e.g. affine, symplectic,complex, Riemannian or pseudo-Riemannian, etc) on T ∗ G , one wonders whether they areequivalent, i.e. if there exists an automorphism of T ∗ G mapping one to the other. Bytaking the values of those structures at the unit of T ∗ G, the problem translates to findingan automorphism of Lie algebra mapping two structures of D . The work within this chaptermay also be seen as a useful tool for the study of such structures. Here are some examplesof problems and framework for further extension and applications of this work. For morediscussions about structures and problems on T ∗ G , one can have a look at [5], [28], [32],[34], [47], [56]. Here, we apply the above results to produce the following examples. Example 2.5.1 (The Affine Lie Algebra of the Real Line) . The 2-dimensional affine Liealgebra G = aff ( R ) is solvable nonnilpotent with Lie bracket [ e , e ] = e , On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ome Possible Applications and Open Problems in some basis ( e , e ). The Lie algebra D = T ∗ G of the cotangent bundle of any Lie groupwith Lie algebra G , has a basis ( e , e , e , e ) with Lie bracket [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , where e := e ∗ and e := e ∗ . This is the semi-direct product R e ⋉ H of the HeisenbergLie algebra H = span ( e , e , e ) with the line R e , where e acts on H by the restrictionof the derivation of ad e . The Lie algebra der ( D ) has a basis ( φ , φ , φ , φ , φ ) where φ ( e ) = e , φ ( e ) = − e , φ ( e ) = e ,φ ( e ) = − e , φ ( e ) = e , φ ( e ) = − e ,φ ( e ) = e , φ ( e ) = e , φ ( e ) = e , the remaining vectors φ i ( e j ) being zero, so that the Lie brackets are [ φ , φ ] = φ , [ φ , φ ] = φ , [ φ , φ ] = φ , [ φ , φ ] = φ . This is the semi-direct product R ⋉ R of the Abelian Lie algebras R = span R ( φ , φ , φ ) and R = span R ( φ , φ ) . It has a contact structure (this is the Lie algebra number 18, for p = q = 1 , in the list of Section 5.2. Example 2.5.2 (The Lie Algebra of the Group SO (3) of Rotations) . Consider the Liealgebra so (3) = span ( e , e , e ) with [ e , e ] = − e , [ e , e ] = e , [ e , e ] = − e . The Lie algebra D = T ∗ G has ( e , e , e , e , e , e ) with Lie bracket [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e , where e i = e ∗ i , i = 1 , , . The Lie algebra der ( D ) is spanned by the elements φ , φ , φ , φ , φ , φ , φ , where φ ( e ) = − e , φ ( e ) = e , φ ( e ) = − e ,φ ( e ) = e , φ ( e ) = − e , φ ( e ) = e ,φ ( e ) = − e , φ ( e ) = e , φ ( e ) = − e ,φ ( e ) = e , φ ( e ) = − e , φ ( e ) = e ,φ ( e ) = − e , φ ( e ) = e , φ ( e ) = − e ,φ ( e ) = e , φ ( e ) = e , φ ( e ) = e ,φ ( e ) = e , φ ( e ) = − e , φ ( e ) = e , so that we have the following Lie brackets [ φ , φ ] = − φ , [ φ , φ ] = φ , [ φ , φ ] = − φ , [ φ , φ ] = φ , [ φ , φ ] = − φ , [ φ , φ ] = φ , [ φ , φ ] = − φ , [ φ , φ ] = − φ , [ φ , φ ] = φ , [ φ , φ ] = − φ , [ φ , φ ] = − φ , [ φ , φ ] = φ . (2.65)This is the Lie algebra der ( D ) = so (3) ⋉ G id , where so (3) = span ( φ , φ , φ ) and G id is thesemi-direct product G id = R φ ⋉ R of the abelian Lie algebras R = span ( φ , φ , φ ) and R φ obtained by letting φ acts as the identity map on R . Thus der ( D ) is also a contactLie algebra, as it is the Lie algebra number 4 of Section 5.3 in [26]. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ome Possible Applications and Open Problems (The Lie Algebra of the Group SL (2 , R ) of Spacial Linear Group) . TheLie algebra G = sl (2 , R ) of SL (2 , R ) has a basis ( e , e , e ) in which its Lie bracket reads [ e , e ] = − e , [ e , e ] = 2 e , [ e , e ] = − e (2.66)Set e ∗ =: e , e ∗ =: e , e ∗ = e , the Lie bracket of D := T ∗ G in the basis ( e , e , e , e , e , e ) is given by [ e , e ] = − e , [ e , e ] = 2 e , [ e , e ] = − e , [ e , e ] = 2 e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = 2 e . (2.67)The Lie algebra der ( D ) is -dimensional. It has a basis ( φ , φ , φ , φ , φ , φ , φ ) , where φ := − e + e + e − e , φ := e + e + e ,φ := − e + 2 e − e + e , φ := e − e + 2 e − e ,φ := − e + e , φ := e − e ,φ := e − e (2.68)Hence, the Lie bracket of der ( D ) reads [ φ , φ ] = φ , [ φ , φ ] = − φ , [ φ , φ ] = φ , [ φ , φ ] = − φ , [ φ , φ ] = φ , [ φ , φ ] = φ , [ φ , φ ] = φ , [ φ , φ ] = 2 φ , [ φ , φ ] = − φ , [ φ , φ ] = − φ , [ φ , φ ] = − φ , [ φ , φ ] = − φ . (2.69)One realizes that der ( D ) = sl (2 , R ) ⋉ G id , where sl (2 , R ) = span ( φ , φ , φ ) and as above, G id is the semi-direct product G id = R φ ⋉ R of the Abelian Lie algebras R = span ( φ , φ , φ ) and R φ obtained by letting φ act as the identity map on R . Again, der ( D ) is also acontact Lie algebra, with η := sφ ∗ + tφ ∗ as an example of a contact form, where s, t ∈ R − { } .On the Lie algebra D = sl (2 , R ) ⋉ sl (2 , R ) ∗ , consider the following two forms ϕ (( x, f ) , ( y, g )) = f ( y ) + g ( y ) (2.70) ϕ (( x, f ) , ( y, g )) = f ( y ) + g ( y ) + K ( x, y ) (2.71)where K stands for the Killing form of sl (2 , R ) . The matrix of ϕ has two eigenvalues, − and +1 , both of multiplicty . Therefore ϕ is of signature (3 , . Now the matrix of ϕ hasfour eigenvalues : − √ , − − √ , − √ , √ , − √ , √ . The three firsteigenvalues are less than zero and the three last ones are positive. Hence the signature of ϕ is (3 , , too. It is now straightforward to check thatder ( D ) = ad D ⊕ R φ (2.72)where ad D := span ( φ , φ , φ , φ , φ , φ ) is the space of inner dérivations of D , φ being, upto multiplication by a scalar, the unique exterior derivation of D .Let us look at the restrictions of ϕ and ϕ to the Levi subalgebra sl (2 , R ) of D .Since the restriction of ϕ to sl (2 , R ) is degenerate, so is the restriction of ϕ to the On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ome Possible Applications and Open Problems image exp( φ i )( sl (2 , R )) of sl (2 , R ) under the standard exponential map of any of the innerderivations φ i , i = 1 , , , , , , of D . That is because special automorphisms exp( φ i ) , i = 1 , , , , , , preserve the Levi subalgebra sl (2 , R ) of D . Suppose that there exists aspecial automorphism exp( φ i ) , i = 1 , , , , , , mapping ϕ to ϕ , i.e. ϕ (cid:16) exp( φ i )( x, f ) , exp( φ i )( y, g ) (cid:17) = ϕ (cid:16) ( x, f ) , ( y, g ) (cid:17) (2.73)for all x, y in sl (2 , R ) and all f, g in sl (2 , R ) ∗ . But the restriction of ϕ to exp( φ i )( sl (2 , R )) isdegenerate while the restriction of ϕ to sl (2 , R ) is equal to the Killing form K of sl (2 , R ) ,which is not degenerate. Hence, ϕ and ϕ are not homothetic via special automorphismsof D .Now, what about exp( φ ) ? One has exp( φ ) = e + e + e + eφ , where e := exp(1) .For any elements ( x, f ) and ( y, g ) of D , we have, ϕ (cid:16) exp( φ )( x, f ) , exp( φ )( y, g ) (cid:17) = ϕ (cid:16) ( x, ef ) , ( y, eg ) (cid:17) = eϕ (cid:16) ( x, f ) , ( y, g ) (cid:17) + K ( x, y ) = ϕ (cid:16) ( x, f ) , ( y, g ) (cid:17) (2.74)Hence, the automorphism exp( φ ) too do not map ϕ to ϕ . As discussed in Section 2.2.1, the Lie group T ∗ G possesses bi-invariant pseudo-Rieman - -nian metrics.Among others, one of the open problems in [60], is the question as to whether, giventwo bi-invariant pseudo-Riemannian metrics µ and µ on a Lie group ˜ G , the two pseudo-Riemannian manifolds ( ˜ G, µ ) and ( ˜ G, µ ) are homothetic via an automorphism of ˜ G. Ifthis is the case, we say that µ and µ are isomorphic or equivalent.When ˜ G = T ∗ G, the question as to how many non-isomorphic bi-invariant pseudo-Riemannian metrics can exist on T ∗ G is still open, in the general case. For example,suppose G itself has a bi-invariant Riemannian or pseudo-Riemannian metric µ and let µ stand again for the corresponding adjoint-invariant metric in the Lie algebra G of G. Then µ induces a new adjoint-invariant metric h , i µ on D = Lie ( T ∗ G ) , with h ( x, f ) , ( y, g ) i µ := h ( x, f ) , ( y, g ) i + µ ( x, y ) , (2.75)for all x, y in G and all f, g in G ∗ , where h , i on the right hand side is the duality pairing h ( x, f ) , ( y, g ) i = f ( y ) + g ( x ) . In some cases (see Section 2.5.3), the two metrics can evenhappen to have the same index, but are still not isomorphic via an automorphism of ˜ G . If ˜ µ is a bilinear symmetric form on G ∗ satisfying ˜ µ ( ad ∗ x f, g ) = 0 , for every x in G and every f, g in G ∗ , then h ( x, f ) , ( y, g ) i ˜ µ := h ( x, f ) , ( y, g ) i + ˜ µ ( f, g ) , (2.76)for all x, y in G , f, g in G ∗ , is also another adjoint-invariant metric on D . Now in some cases,nonzero such bilinear forms ˜ µ may not exist, as is the case when one of the coadjoint orbits On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ome Possible Applications and Open Problems of G, is an open subset of G ∗ , or equivalently, when G has a left invariant exact symplecticstructure.More generally, these equivalence questions can also be simply extended to all left (resp.right) invariant Riemannian or pseudo-Riemannian structures on cotangent bundles of Liegroups. The classification questions of double Lie algebras, Manin pairs, Lagrangian subalgebras,... arising from Poisson-Lie groups, are still open problems [32], [55]. In [32], Lagrangiansubalgebras of double Lie algebras are used as the main tool for classifying the so-calledPoisson Homogeneous spaces of Poisson-Lie groups. A type of local action of those La-grangian subalgebras is also used to describe symplectic foliations of Poisson Homogeneousspaces of Poisson-Lie groups in [32], [27].It would be interesting to extend the results within this chapter to double Lie algebrasof general Poisson-Lie groups. It is hard to get a common substantial description valid forthe group of automorphism of the double Lie algebras of all possible Poisson-Lie structuresin a given Lie group. This is due to the diversity of Poisson-Lie structures that can coexistin the same Lie group.Among other things, the description of the group of automorphisms of the double Liealgebra of a Poisson-Lie structure is a big step forward towards solving very interestingand hard problems such as:- the classification of Manin triples ([55]);- the classification of Poisson homogeneous spaces of a Lie groups ([27], [55]);- a full description and understanding of the foliations of Poisson homogeneous spacesof Poisson Lie groups. The leaves of such foliations trap the trajectories, under aHamiltonian flow, passing through all its points. Hence, from their knowledge, onegets a great deal of information on Hamiltonian systems.- etc. T ∗ G In certain cases, T ∗ G possesses left invariant affine connections, that is, left invariant zerocurvature and torsion free linear connections. Here, the classification problem involvesAut( T ∗ G ) as follows. The group Aut( T ∗ G ) acts on the space of left invariant affine connec-tions on T ∗ G , the orbit of each connection being the set of equivalent (isomorphic) ones.Recall that among other results in [28], the authors proved that when G has an invertiblesolution of the Classical Yang-Baxter Equation (or equivalently a left invariant symplecticstructure), then T ∗ G has a left invariant affine connection ∇ and a complex structure J such that ∇ J = 0 . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 hapter Three Prederivations of Lie Algebras ofCotangent Bundles of Lie Groups Contents T ∗ G . . . . . . . . . . . . . . . . . . . . . 513.4 Orthogonal Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 613.5 Possible Applications and Examples . . . . . . . . . . . . . . . 70 In the sense of Felix Klein ([43]), studying the geometry of a "universe" is studying itsinvariant structures under the action of a suitable Lie group. In semi-Riemannian geometry,one of the suitable group used in this task is the group of isometries of pseudo-Riemannianmetrics. So it seems reasonable to well known isometries of pseudo-Riemannian metrics.Among tools used, for instance in the case of bi-invariant (or orthogonal) Lie groups,there are prederivations of Lie algebras. Müller ([64]) gives an algebraic description of thegroup I ( G, µ ) of isometries of a connected orthogonal Lie group ( G, µ ) . He proves thatif ( G, µ ) is a connected and simply-connected orthogonal Lie group with Lie algebra G ,then the stabilizer of the identity element of G in I ( G, µ ) is isomorphic to the group ofpreautomorphisms of G which preserve the non-degenerate bilinear form induced by µ on G and whose Lie algebra is the whole set of skew-symmetric prederivations of G . In [9], Bajostudies the algebra of prederivations and skew-symmetric prederivations of a direct sum ofLie algebras and this study allows him to generalize some results in [6], [7] and in [72].Prederivations also present an interest in the purely algebraic point of view. As wellas Jacobson ([40]) proves that a Lie algebra admitting a non-singular derivation is neces-sarily nilpotent, the author quoted above establishes in [8] that a Lie algebra possessing anon-singular prederivation is necessarily nilpotent. Prederivations are also useful tools forconstruction of affine structures on Lie algebras (see Section 3.5.2). On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ntroduction In this chapter, we deal with prederivations of the Lie algebra of the cotangent bundleof a Lie group; which Lie algebra appears as the semi-direct sum G ⋉ A of a Lie algebra G and an Abelian Lie algebra A . In the sequel, we will take G to be the Lie algebra of a Liegroup G , A = G ∗ , the dual space of G and T ∗ G := G ⋉ G ∗ will be the semi-direct sum ofthe Lie algebra G and the vector space G ∗ via the coadjoint representation.Our aim is to explore the structure of the Lie algebra Pder ( T ∗ G ) of prederivations of T ∗ G . We will have a particular attention to Lie algebras admitting quadratic or orthogonalstructures.The main results within this chapter are the following. Theorem A. A prederivation p : T ∗ G → T ∗ G is defined by : p ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ξ ( f ) (cid:17) for any element ( x, f ) of T ∗ G , where α : G → G is a prederivation of G and β : G → G ∗ , ψ : G ∗ → G and ξ : G ∗ → G ∗ are linear maps satisfying the following four relations : β (cid:0) [ x, [ y, z ]] (cid:1) = − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) + ad ∗ x (cid:16) ad ∗ y (cid:0) β ( z ) (cid:1) − ad ∗ z (cid:0) β ( y ) (cid:1)(cid:17) ψ ◦ ad ∗ [ x,y ] = ad [ x,y ] ◦ ψad ∗ x (cid:16) ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f (cid:17) = 0 , h ξ, ad ∗ [ x,y ] i = ad ∗ ([ α ( x ) ,y ]+[ x,α ( y )]) for every elements x and y of G and any elements f and g in G ∗ . About the structure of the Lie algebra of the Lie algebra Pder ( T ∗ G ) of prederivationsof T ∗ G , we have the Theorem B. Let G be any finite-dimensional Lie group with Lie algebra G . Then the Liealgebra Pder ( T ∗ G ) of prederivations of the Lie algebra T ∗ G of the Lie group T ∗ G decomposesas follows : Pder ( T ∗ G ) = G ′ ⊕ G ′ , where G ′ is a reductive subalgebra of Pder ( T ∗ G ) , that is [ G ′ , G ′ ] ⊂ G ′ and [ G ′ , G ′ ] ⊂ G ′ . Recall that the Lie algebra of the cotangent bundle Lie group of a semi-simple Liegroup is not semi-simple. Any way, as well as Müller proves that any prederivation of asemi-simple Lie algebra is a derivation we prove the following Theorem C. Let G be a semi-simple Lie group with Lie algebra G . Then every prederivationof the Lie algebra T ∗ G of the cotangent bundle Lie group T ∗ G of G is a derivation. The chapter contains five ( ) sections. In Section 3.2 is explained the link betweenprederivations and isometries of bi-invariant metrics on Lie groups. In Section 3.3 we study On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reliminaries the structure of the Lie algebra Pder ( T ∗ G ) of prederivations of the Lie algebra T ∗ G of thecotangent bundle T ∗ G of a Lie group G with Lie G . The particular case where the Liegroup G possesses a bi-invariant metric is studied in Section 3.4. In Section 3.5 we giveexamples and some possible applications. Definition 3.2.1. Let G be a Lie algebra. A bijective endomorphism P : G → G such that P (cid:0)(cid:2) x, [ y, z ] (cid:3)(cid:1) = (cid:2) P ( x ) , [ P ( y ) , P ( z )] (cid:3) , (3.1) for all x, y, z in G , is called a preautomorphism of G . The set of all preautomorphisms of G forms a Lie group (see [64]) which we note byPaut ( T ∗ G ) . Its Lie algebra is the subset of the set G l ( G ) of endomorphisms of G consistingof elements p which satisfy the following relation. p (cid:0)(cid:2) x, [ y, z ] (cid:3)(cid:1) = (cid:2) p ( x ) , [ y, z ] (cid:3) + (cid:2) x, [ p ( y ) , z ] (cid:3) + (cid:2) x, [ y, p ( z )] (cid:3) (3.2)for every elements x, y, z of G .One can easily convince himself that any derivation of a Lie algebra is a prederivation.Furthermore, there exists a class of Lie algebras that are such that any prederivation isa derivation. Semi-simple Lie algebras belong to that class of algebras (see [64]). We willprove in Section 3.4.2 that the Lie algebras (which are not semi-simple) of the cotangentLie groups of semi-simple Lie groups are also members of this class.As well as Jacobson ([40]) proves that a Lie algebra admitting a non-singular derivationis necessarily nilpotent, Bajo ([8]) shows that a Lie algebra which possesses a non-singularprederivation is necessarily nilpotent. The converses of the both results are false since thereare nilpotent Lie algebra that admit only singular derivations and prederivations (see [8]). Müller ([64]) proves that if ( G, µ ) is a connected and simply-connected orthogonal Lie groupwith Lie algebra G , then the isotropy group of the neutral element of G in the group I ( G, µ ) of isometries of ( G, µ ) is isomorphic to the subgroup of GL ( G ) (group of endomorphismsof G ) consisting of preautomorphisms of G which preserve the non-degenerate bilinear forminduced by µ on G and whose Lie algebra is the whole set of skew-symmetric prederivationsof G . See Section 4.2.3 for wider informations. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G T ∗ G T ∗ G Let G be a Lie algebra and let T ∗ G := G ⋉ G ∗ stand for the semi-direct product of G withits dual via the coadjoint representation. The Lie bracket of T ∗ G is given by [( x, f ) , ( y, g )] := (cid:16) [ x, y ] , ad ∗ x g − ad ∗ y f (cid:17) , (3.3)for any elements ( x, f ) and ( y, g ) of T ∗ G .Let p : T ∗ G → T ∗ G be a prederivation of T ∗ G and set p ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ξ ( f ) (cid:17) , (3.4)where α : G → G , ψ : G ∗ → G , β : G → G ∗ and ξ : G ∗ → G ∗ are linear maps.Let x, y, z be elements of G . We have : p (cid:0) [ x, [ y, z ]] (cid:1) = α (cid:0) [ x, [ y, z ]] (cid:1)| {z } ∈ G + β (cid:0) [ x, [ y, z ]] (cid:1)| {z } ∈ G ∗ (3.5)On the other way, since p is a prederivation, then p (cid:0) [ x, [ y, z ]] (cid:1) = h p ( x ) , [ y, z ] i + h x, [ p ( y ) , z ] i + h x, [ y, p ( z )] i = h α ( x ) + β ( x ) , [ y, z ] i + h x, [ α ( y ) + β ( y ) , z ] i + h x, [ y, α ( z ) + β ( z )] i = (cid:16)h α ( x ) , [ y, z ] i − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1)(cid:17) + h x, [ α ( y ) , z ] − ad ∗ z (cid:0) β ( y ) (cid:1)i + h x, [ y, α ( z )] + ad ∗ y (cid:0) β ( z ) (cid:1)i = (cid:16)h α ( x ) , [ y, z ] i − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1)(cid:17) + (cid:16)h x, [ α ( y ) , z ] i − ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1)(cid:17) + (cid:16)h x, [ y, α ( z )] i + ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1)(cid:17) = (cid:16)h α ( x ) , [ y, z ] i + h x, [ α ( y ) , z ] i + h x, [ y, α ( z )] i(cid:17)| {z } ∈ G + (cid:16) − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) − ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1)(cid:17)| {z } ∈ G ∗ . (3.6)From relations (3.5) and (3.6) we have : α (cid:16)h x, [ y, z ] i(cid:17) = h α ( x ) , [ y, z ] i + h x, [ α ( y ) , z ] i + h x, [ y, α ( z )] i . (3.7)for any x, y, z in G ; that is α is a prederivation of G . Relations (3.5) and (3.6) also give β (cid:16)h x, [ y, z ] i(cid:17) = − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) − ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1) . (3.8) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G Now let x be an element of G and f, g be in G ∗ . We have p ([ x, [ f, g ]]) = p (0) = 0 . (3.9)We also have p ([ x, [ f, g ]]) = h p ( x ) , [ f, g ] i + h x, [ p ( f ) , g ] i + h x, [ f, p ( g )] i = 0 + h x, [ ψ ( f ) + ξ ( f ) , g ] i + [ x, [ f, ψ ( g ) + ξ ( g )] i = [ x, ad ∗ ψ ( f ) g ] + [ x, − ad ∗ ψ ( g ) f ]= [ x, ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f ]= ad ∗ x (cid:16) ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f (cid:17) . (3.10)From (3.9) and (3.10) it comes that ad ∗ x (cid:16) ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f (cid:17) = 0 , (3.11)for every x in G and every f, g in G ∗ . That is ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f belongs the centralizer Z T ∗ G ( G ) of G in T ∗ G , for any f and g in G ∗ .Let us now consider the following case. Let x and y be in G and f be in G ∗ . p (cid:16)h x, [ y, f ] i(cid:17) = ψ (cid:16)h x, [ y, f ] i(cid:17)| {z } ∈ G + ξ (cid:16)h x, [ y, f ] i(cid:17)| {z } ∈ G ∗ (3.12)Since p is a prederivation, one has p (cid:16)h x, [ y, f ] i(cid:17) = h p ( x ) , [ y, f ] i + h x, [ p ( y ) , f ] i + h x, [ y, p ( f )] i = h α ( x ) + β ( x ) , [ y, f ] i + h x, [ α ( y ) + β ( y ) , f ] i + h x, [ y, ψ ( f ) + ξ ( f )] i = h α ( x ) + β ( x ) , ad ∗ y f i + h x, ad ∗ α ( y ) f i + h x, [ y, ψ ( f )] + ad ∗ y (cid:0) ξ ( f ) (cid:1)i = ad ∗ α ( x ) ◦ ad ∗ y ( f ) + ad ∗ x ◦ ad ∗ α ( y ) ( f ) + (cid:2) x, [ y, ψ ( f )] (cid:3) + ad ∗ x ◦ ad ∗ y (cid:0) ξ ( f ) (cid:1) = [ x, [ y, ψ ( f )] | {z } ∈ G + (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) ( f ) | {z } ∈ G ∗ (3.13)On one hand, relations (3.12) and (3.13) imply ψ (cid:16)h x, [ y, f ] i(cid:17) = [ x, [ y, ψ ( f )] ψ (cid:16) ad ∗ x ◦ ad ∗ y f (cid:17) = ad x ◦ ad y ◦ ψ ( f ) , that is ψ ◦ ad ∗ x ◦ ad ∗ y = ad x ◦ ad y ◦ ψ, (3.14)for any x and y in G . On the second hand relations (3.12) and (3.13) give ξ (cid:16)h x, [ y, f ] i(cid:17) = (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) ( f ) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G ξ ◦ ad ∗ x ◦ ad ∗ y ( f ) = (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) ( f ) (cid:16) ξ ◦ ad ∗ x ◦ ad ∗ y − ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) ( f ) = (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) (cid:17) ( f ) It comes that [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) (3.15)Let us summarize in the Theorem 3.3.1. A prederivation p : T ∗ G → T ∗ G is defined by : p ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ξ ( f ) (cid:17) (3.16) for any element ( x, f ) of T ∗ G , where • α : G → G is a prederivation of G and • β : G → G ∗ , ψ : G ∗ → G and ξ : G ∗ → G ∗ are linear maps satisfying the following fourrelations : β (cid:0) [ x, [ y, z ]] (cid:1) = − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) + ad ∗ x (cid:16) ad ∗ y (cid:0) β ( z ) (cid:1) − ad ∗ z (cid:0) β ( y ) (cid:1)(cid:17) (3.17) ψ ◦ ad ∗ x ◦ ad ∗ y = ad x ◦ ad y ◦ ψ, (3.18) ad ∗ x (cid:16) ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f (cid:17) = 0 , (3.19) [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) , (3.20) for every elements x and y of G and any elements f and g in G ∗ . ( T ∗ G ) Let us introduce the following notations :1. Pder ( T ∗ G ) stands for the space of prederivations of T ∗ G ;2. Pder ( G ) represents for the space of prederivations of G ;3. Q ′ is the space of linear maps β : G → G ∗ satisfying relation (3.17) ;4. E ′ is the space of linear maps ξ : G ∗ → G ∗ such that [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) , for some prederivation α of G and any elements x and y of G .5. Ψ ′ stands for the space of linear maps ψ : G ∗ → G satisfying (3.18) and (3.19).6. G ′ stands for the space of maps φ α,ξ : T ∗ G → T ∗ G , ( x, f ) (cid:0) α ( x ) , ξ ( f ) (cid:1) , where α isin Pder ( G ) , ξ in E ′ and [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) , for all x, y in G ;7. G ′ := Ψ ′ ⊕ Q ′ (direct sum of vector spaces). On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G The space E ′ is a Lie algebra. Precisely, if ξ , ξ in E ′ satisfy (cid:2) ξ , ad ∗ x ◦ ad ∗ y (cid:3) = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) (cid:2) ξ , ad ∗ x ◦ ad ∗ y (cid:3) = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) for all x , y in G and some α , α in Pder ( G ) , then [ ξ , ξ ] belongs to E ′ and satisfies h [ ξ , ξ ] , ad ∗ x ◦ ad ∗ y i = ad ∗ [ α ,α ]( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ [ α ,α ]( y ) , for all elements x, y of G .Proof. Consider ξ and ξ as in Lemma 3.3.1. We have, h [ ξ , ξ ] , ad ∗ x ◦ ad ∗ y i = (cid:2) ξ ◦ ξ − ξ ◦ ξ , ad ∗ x ◦ ad ∗ y (cid:3) = (cid:2) ξ ◦ ξ , ad ∗ x ◦ ad ∗ y (cid:3) − (cid:2) ξ ◦ ξ , ad ∗ x ◦ ad ∗ y (cid:3) (3.21) (cid:2) ξ ◦ ξ , ad ∗ x ◦ ad ∗ y (cid:3) = ( ξ ◦ ξ ) ◦ ad ∗ x ◦ ad ∗ y − ad ∗ x ◦ ad ∗ y ◦ ( ξ ◦ ξ )= ξ (cid:16) ξ ◦ ad ∗ x ◦ ad ∗ y (cid:17) − (cid:16) ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) ◦ ξ = (cid:16) ξ ◦ ad ∗ α ( x ) ◦ ad ∗ y + ξ ◦ ad ∗ x ◦ ad ∗ α ( y ) + ξ ◦ ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) − (cid:16) ξ ◦ ad ∗ x ◦ ad ∗ y − ad ∗ α ( x ) ◦ ad ∗ y − ad ∗ x ◦ ad ∗ α ( y ) (cid:17) ◦ ξ (3.22)Hence, (cid:2) ξ ◦ ξ , ad ∗ x ◦ ad ∗ y (cid:3) = ξ ◦ ad ∗ α ( x ) ◦ ad ∗ y + ξ ◦ ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ α ( x ) ◦ ad ∗ y ◦ ξ + ad ∗ x ◦ ad ∗ α ( y ) ◦ ξ (3.23)We also have : (cid:2) ξ ◦ ξ , ad ∗ x ◦ ad ∗ y (cid:3) = ξ ◦ ad ∗ α ( x ) ◦ ad ∗ y + ξ ◦ ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ α ( x ) ◦ ad ∗ y ◦ ξ + ad ∗ x ◦ ad ∗ α ( y ) ◦ ξ (3.24)Now we have h [ ξ , ξ ] , ad ∗ x ◦ ad ∗ y i = (cid:16) ξ ◦ ad ∗ α ( x ) ◦ ad ∗ y + ξ ◦ ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ α ( x ) ◦ ad ∗ y ◦ ξ + ad ∗ x ◦ ad ∗ α ( y ) ◦ ξ (cid:17) − (cid:16) ξ ◦ ad ∗ α ( x ) ◦ ad ∗ y + ξ ◦ ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ α ( x ) ◦ ad ∗ y ◦ ξ + ad ∗ x ◦ ad ∗ α ( y ) ◦ ξ (cid:17) = (cid:16) ξ ◦ ad ∗ α ( x ) ◦ ad ∗ y − ad ∗ α ( x ) ◦ ad ∗ y ◦ ξ (cid:17) + (cid:16) ξ ◦ ad ∗ x ◦ ad ∗ α ( y ) − ad ∗ x ◦ ad ∗ α ( y ) ◦ ξ (cid:17) + (cid:16) ad ∗ α ( x ) ◦ ad ∗ y ◦ ξ − ξ ◦ ad ∗ α ( x ) ◦ ad ∗ y (cid:17) + (cid:16) ad ∗ x ◦ ad ∗ α ( y ) ◦ ξ − ξ ◦ ad ∗ x ◦ ad ∗ α ( y ) (cid:17) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G = (cid:2) ξ , ad ∗ α ( x ) ◦ ad ∗ y (cid:3) + (cid:2) ξ , ad ∗ x ◦ ad ∗ α ( y ) (cid:3) − (cid:2) ξ , ad ∗ α ( x ) ◦ ad ∗ y (cid:3) − (cid:2) ξ , ad ∗ x ◦ ad ∗ α ( y ) (cid:3) = (cid:16) ad ∗ α ◦ α ( x ) ◦ ad ∗ y + ad ∗ α ( x ) ◦ ad ∗ α ( y ) (cid:17) + (cid:16) ad ∗ α ( x ) ◦ ad ∗ α ( y ) + ad ∗ x ◦ ad ∗ α ◦ α ( y ) (cid:17) − (cid:16) ad ∗ α ◦ α ( x ) ◦ ad ∗ y + ad ∗ α ( x ) ◦ ad ∗ α ( y ) (cid:17) − (cid:16) ad ∗ α ( x ) ◦ ad ∗ α ( y ) + ad ∗ x ◦ ad ∗ α ◦ α ( y ) (cid:17) . We then have h [ ξ , ξ ] , ad ∗ x ◦ ad ∗ y i = ad ∗ [ α ,α ]( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ [ α ,α ]( y ) . Lemma 3.3.2. The space G ′ is a Lie subalgebra of the Lie algebra Pder ( T ∗ G ) .Proof. Let φ α ,ξ and φ α ,ξ be two elements of G ′ . (cid:16) φ α ,ξ ◦ φ α ,ξ (cid:17) ( x, f ) = φ α ,ξ (cid:16) α ( x ) , ξ ( f ) (cid:17) = (cid:16) α ◦ α ( x ) , ξ ◦ ξ ( f ) (cid:17) We then have (cid:16) φ α ,ξ ◦ φ α ,ξ (cid:17) = (cid:16) α ◦ α , ξ ◦ ξ (cid:17) . By the same way, we have (cid:16) φ α ,ξ ◦ φ α ,ξ (cid:17) = (cid:16) α ◦ α , ξ ◦ ξ (cid:17) . Hence, [ φ α ,ξ , φ α ,ξ ] = (cid:16) [ α , α ] , [ ξ , ξ ] (cid:17) ∈ G ′ . Lemma 3.3.3. [ G ′ , Ψ ′ ] ⊂ Ψ ′ and [ G ′ , Q ′ ] ⊂ Q ′ . Hence, [ G ′ , G ′ ] ⊂ G ′ .Proof. Let φ α,ξ and φ ψ, be elements of G ′ and Ψ ′ respectively. We have φ α,ξ ◦ φ ψ, ( x, f ) = φ α,ξ (cid:16) ψ ( f ) , (cid:17) = (cid:16) α ◦ ψ ( f ) , (cid:17) .φ ψ, ◦ φ α,ξ ( x, f ) = φ ψ, (cid:16) α ( x ) , ξ ( f ) (cid:17) = (cid:16) ψ ◦ ξ ( f ) , (cid:17) . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G Then, [ φ α,ξ , φ ψ,β ]( x, f ) = (cid:16) ( α ◦ ψ − ψ ◦ ξ )( f ) , (cid:17) . Let us show that ( α ◦ ψ − ψ ◦ ξ ) belongs to Ψ ′ . For any elements x and y in G , we have ( α ◦ ψ ) ◦ ad ∗ x ◦ ad ∗ y = α ◦ ( ψ ◦ ad ∗ x ◦ ad ∗ y )= α ◦ ( ad x ◦ ad y ◦ ψ )= ( α ◦ ad x ◦ ad y ) ◦ ψ. Let z be an element of G . We have ( α ◦ ad x ◦ ad y )( z ) = α ([ x, [ y, z ]])= h α ( x ) , [ y, z ] i + h x, [ α ( y ) , z ] i + h x, [ y, α ( z )] i = (cid:16) ad α ( x ) ◦ ad y + ad x ◦ ad α ( y ) + ad x ◦ ad y ◦ α (cid:17) ( z ) . Then α ◦ ad x ◦ ad y = ad α ( x ) ◦ ad y + ad x ◦ ad α ( y ) + ad x ◦ ad y ◦ α. It comes that : ( α ◦ ψ ) ◦ ad ∗ x ◦ ad ∗ y = ad α ( x ) ◦ ad y ◦ ψ + ad x ◦ ad α ( y ) ◦ ψ + ad x ◦ ad y ◦ α ◦ ψ. (3.25)Now, what about ( ψ ◦ ξ ) ◦ ad ∗ x ◦ ad ∗ y ? ( ψ ◦ ξ ) ◦ ad ∗ x ◦ ad ∗ y = ψ ◦ ( ξ ◦ ad ∗ x ◦ ad ∗ y )= ψ ◦ (cid:16) [ ξ, ad ∗ x ◦ ad ∗ y ] + ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) = ψ ◦ (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) + ad ∗ x ◦ ad ∗ y ◦ ξ (cid:17) . Hence, ( ψ ◦ ξ ) ◦ ad ∗ x ◦ ad ∗ y = ψ ◦ ad ∗ α ( x ) ◦ ad ∗ y + ψ ◦ ad ∗ x ◦ ad ∗ α ( y ) + ψ ◦ ad ∗ x ◦ ad ∗ y ◦ ξ (3.26)From (3.25) and (3.26), we have : ( α ◦ ψ − ψ ◦ ξ ) ◦ ad ∗ x ◦ ad ∗ y = ( ad α ( x ) ◦ ad y ◦ ψ + ad x ◦ ad α ( y ) ◦ ψ + ad x ◦ ad y ◦ α ◦ ψ ) − ( ψ ◦ ad ∗ α ( x ) ◦ ad ∗ y + ψ ◦ ad ∗ x ◦ ad ∗ α ( y ) + ψ ◦ ad ∗ x ◦ ad ∗ y ◦ ξ ) . It comes that ( α ◦ ψ − ψ ◦ ξ ) satisfies (3.18) as it verifies ( α ◦ ψ − ψ ◦ ξ ) ◦ ad ∗ x ◦ ad ∗ y = ad x ◦ ad y ◦ ( α ◦ ψ − ψ ◦ ξ ) . Let now f and g be elements of G ∗ . Γ := ad ∗ ( α ◦ ψ − ψ ◦ ξ )( f ) g − ad ∗ ( α ◦ ψ − ψ ◦ ξ )( g ) f = ad ∗ α ◦ ψ ( f ) g − ad ∗ ψ ◦ ξ ( f ) g − ad ∗ α ◦ ψ ( g ) f + ad ∗ ψ ◦ ξ ( g ) f For any x in G , one has the following ad ∗ x ◦ ad ∗ α ◦ ψ ( f ) = [ ξ, ad ∗ x ◦ ad ∗ ψ ( f ) ] − ad ∗ α ( x ) ◦ ad ∗ ψ ( f ) ad ∗ x ◦ ad ∗ α ◦ ψ ( g ) = [ ξ, ad ∗ x ◦ ad ∗ ψ ( g ) ] − ad ∗ α ( x ) ◦ ad ∗ ψ ( g ) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G Then ad ∗ x ◦ Γ = ad ∗ x ◦ (cid:16) ad ∗ ( α ◦ ψ − ψ ◦ ξ )( f ) g − ad ∗ ( α ◦ ψ − ψ ◦ ξ )( g ) f (cid:17) = ξ ◦ ad ∗ x ◦ ad ∗ ψ ( f ) ( g ) − ad ∗ x ◦ ad ∗ ψ ( f ) ◦ ξ ( g ) − ad ∗ α ( x ) ◦ ad ∗ ψ ( f ) ( g ) − ξ ◦ ad ∗ x ◦ ad ∗ ψ ( g ) ( f ) + ad ∗ x ◦ ad ∗ ψ ( g ) ◦ ξ ( f ) + ad ∗ α ( x ) ◦ ad ∗ ψ ( g ) ( f ) − ad ∗ x ◦ ad ∗ ψ ◦ ξ ( f ) ( g ) + ad ∗ x ◦ ad ∗ ψ ◦ ξ ( g ) ( f )= ξ (cid:16) ad ∗ x ◦ ad ∗ ψ ( f ) ( g ) − ◦ ad ∗ x ◦ ad ∗ ψ ( g ) ( f ) (cid:17)| {z } =0 , because of (3.19) + ad ∗ α ( x ) (cid:16) ad ∗ ψ ( g ) ( f ) − ad ∗ ψ ( f ) ( g ) (cid:17)| {z } =0 , because of (3.19) + ad ∗ x ◦ ad ∗ ψ ◦ ξ ( g ) f − ad ∗ x ◦ ad ∗ ψ ( ξ ( g )) f | {z } =0 + ad ∗ x ◦ ad ∗ ψ ( ξ ( f )) g − ad ∗ x ◦ ad ∗ ψ ◦ ξ ( f ) g | {z } =0 . Hence, ad ∗ x (cid:16) ad ∗ ( α ◦ ψ − ψ ◦ ξ )( f ) g − ad ∗ ( α ◦ ψ − ψ ◦ ξ )( g ) f (cid:17) = 0 , (3.27)for any f, g in G ∗ and any x in G . We then have shown that ( α ◦ ψ − ψ ◦ ξ ) belongs to Ψ ′ .Hence, [ G ′ , Ψ ′ ] ⊂ Ψ ′ .Now we are going to show that [ G ′ , Q ′ ] ⊂ Q ′ . For this goal, let φ α,ξ and φ ,β be elementsof G ′ and Q ′ respectively. φ α,ξ ◦ φ ,β ( x, f ) = φ α,ξ (cid:16) , β ( x ) (cid:17) = (cid:16) , ξ ◦ β ( x ) (cid:17) (3.28) φ ,β ◦ φ α,ξ ( x, f ) = φ ,β (cid:16) α ( x ) , ξ ( f ) (cid:17) = (cid:16) , β ◦ α ( x ) (cid:17) (3.29)Then, [ φ α,ξ , φ ,β ]( x, f ) = (cid:16) , ( ξ ◦ β − β ◦ α )( x ) (cid:17) (3.30)Does ( ξ ◦ β − β ◦ α ) satisfies relation (3.17) ? ( ξ ◦ β )[ x, [ y, z ]] = ξ (cid:16) − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) − ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1)(cid:17) = − ξ ◦ ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) − ξ ◦ ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ξ ◦ ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1) = − ξ ◦ ad ∗ y ◦ ad ∗ z (cid:0) β ( x ) (cid:1) + ξ ◦ ad ∗ z ◦ ad ∗ y (cid:0) β ( x ) (cid:1) − ξ ◦ ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ξ ◦ ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1) = − [ ξ, ad ∗ y ◦ ad ∗ z ]( β ( x )) − ad ∗ y ◦ ad ∗ z ◦ ξ ( β ( x ))+[ ξ, ad ∗ z ◦ ad ∗ y ]( β ( x )) + ad ∗ z ◦ ad ∗ y ◦ ξ ( β ( x )) − [ ξ, ad ∗ x ◦ ad ∗ z ]( β ( y )) − ad ∗ x ◦ ad ∗ z ◦ ξ ( β ( y ))+[ ξ, ad ∗ x ◦ ad ∗ y ]( β ( z )) + ad ∗ x ◦ ad ∗ y ◦ ξ ( β ( z ))= − ad ∗ α ( y ) ◦ ad ∗ z ( β ( x )) − ad ∗ y ◦ ad ∗ α ( z ) ( β ( x )) − ad ∗ y ◦ ad ∗ z ◦ ξ ( β ( x ))+ ad ∗ α ( z ) ◦ ad ∗ y ( β ( x )) + ad ∗ z ◦ ad ∗ α ( y ) ( β ( x )) + ad ∗ z ◦ ad ∗ y ◦ ξ ( β ( x )) − ad ∗ α ( x ) ◦ ad ∗ z ( β ( y )) − ad ∗ x ◦ ad ∗ α ( z ) ( β ( y )) − ad ∗ x ◦ ad ∗ z ◦ ξ ( β ( y ))+ ad ∗ α ( x ) ◦ ad ∗ y ( β ( z ))+ ad ∗ x ◦ ad ∗ α ( y ) ( β ( z ))+ ad ∗ x ◦ ad ∗ y ◦ ξ ( β ( z )) (3.31) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G ( β ◦ α )[ x, [ y, z ]] = β (cid:16) [ α ( x ) , [ y, z ]] + [ x, [ α ( y ) , z ]] + [ x, [ y, α ( z )]] (cid:17) = − ad ∗ [ y,z ] (cid:0) β ( α ( x )) (cid:1) − ad ∗ α ( x ) ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ad ∗ α ( x ) ◦ ad ∗ y (cid:0) β ( z ) (cid:1) − ad ∗ [ α ( y ) ,z ] (cid:0) β ( x ) (cid:1) − ad ∗ x ◦ ad ∗ z (cid:0) β ( α ( y )) (cid:1) + ad ∗ x ◦ ad ∗ α ( y ) (cid:0) β ( z ) (cid:1) − ad ∗ [ y,α ( z )] (cid:0) β ( x ) (cid:1) − ad ∗ x ◦ ad ∗ α ( z ) (cid:0) β ( y ) (cid:1) + ad ∗ x ◦ ad ∗ y (cid:0) β ( α ( z )) (cid:1) (3.32)We then have ( ξ ◦ β − β ◦ α )[ x, [ y, z ]] = − ad ∗ [ y,z ] (cid:16) ( ξ ◦ β )( x ) (cid:17) − ad ∗ [ α ( y ) ,z ] ( β ( x )) − ad ∗ [ y,α ( z )] ( β ( x )) − ad ∗ x ◦ ad ∗ z (cid:16) ( ξ ◦ β )( y ) (cid:17) + ad ∗ x ◦ ad ∗ y (cid:16) ( ξ ◦ β )( z ) (cid:17) + ad ∗ [ y,z ] (cid:16) ( β ◦ α )( x ) (cid:17) + ad ∗ [ α ( y ) ,z ] ( β ( x ))+ ad ∗ x ◦ ad ∗ z (cid:16) ( β ◦ α )( y ) (cid:17) + ad ∗ [ y,α ( z )] ( β ( x )) − ad ∗ x ◦ ad ∗ y (cid:16) ( β ◦ α )( z ) (cid:17) = − ad ∗ [ y,z ] (cid:16) ( ξ ◦ β − β ◦ α )( x ) (cid:17) − ad ∗ x ◦ ad ∗ z (cid:16) ( ξ ◦ β − β ◦ α )( y ) (cid:17) + ad ∗ x ◦ ad ∗ y (cid:16) ( ξ ◦ β − β ◦ α )( z ) (cid:17) (3.33)That is ( ξ ◦ β − β ◦ α ) satisfies relation (3.17) and then [ G ′ , Q ′ ] ⊂ Q ′ . It is now clear that [ G ′ , G ′ ] ⊂ G ′ .We summarize the Lemmas above in the Theorem 3.3.2. Let G be any finite-dimensional Lie algebra. Then the Lie algebra ofprederivations of T ∗ G decomposes as follows : Pder ( T ∗ G ) = G ′ ⊕ G ′ , where G ′ is a reductivesubalgebra of Pder ( T ∗ G ) , that is [ G ′ , G ′ ] ⊂ G ′ and [ G ′ , G ′ ] ⊂ G ′ . Remark 3.3.1. Pder ( T ∗ G ) is not a symmetric space as is der ( T ∗ G ) (see Theorem 2.3.2)since [ G ′ , G ′ ] is not a subset of G ′ . Precisely, let φ ψ ,β and φ ψ ,β be two elements of G ′ .Then,1. [ φ ψ ,β , φ ψ ,β ] = (cid:16) ( ψ ◦ β − ψ ◦ β ) , ( β ◦ ψ − β ◦ ψ ) (cid:17) ;2. [ φ ψ ,β , φ ψ ,β ] do not belong to G ′ even ( ψ ◦ β − ψ ◦ β ) is a prederivation of G .3. ( β ◦ ψ − β ◦ ψ ) is not linked to ( ψ ◦ β − ψ ◦ β ) by (3.20). Indeed, let φ ψ ,β and φ ψ ,β be two elements of G ′ . ( φ ψ ,β ◦ φ ψ ,β )( x, f ) = φ ψ ,β (cid:16) ψ ( f ) , β ( x ) (cid:17) = (cid:16) ψ ◦ β ( x ) , β ◦ ψ ( f ) (cid:17) (3.34)By the same way ( φ ψ ,β ◦ φ ψ ,β )( x, f ) = (cid:16) ψ ◦ β ( x ) , β ◦ ψ ( f ) (cid:17) (3.35) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G It comes that [ φ ψ ,β , φ ψ ,β ]( x, f ) = (cid:16) ( ψ ◦ β − ψ ◦ β )( x ) , ( β ◦ ψ − β ◦ ψ )( f ) (cid:17) (3.36)Let us see if ( ψ ◦ β − ψ ◦ β ) is a prederivation of G . ( ψ ◦ β )[ x, [ y, z ]] = ψ (cid:16) − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) − ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1)(cid:17) = − ψ ◦ ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) − ψ ◦ ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1) + ψ ◦ ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1) = − ad y ◦ ad z ◦ ψ (cid:0) β ( x ) (cid:1) + ad z ◦ ad y ◦ ψ (cid:0) β ( x ) (cid:1) − ad x ◦ ad z ◦ ψ (cid:0) β ( y ) (cid:1) + ad x ◦ ad y ◦ ψ (cid:0) β ( z ) (cid:1) = − h y, [ z, ψ ◦ β ( x )] i + h z, [ y, ψ ◦ β ( x )] i − h x, [ z, ψ ◦ β ( y )] i + h x, [ y, ψ ◦ β ( z )] i (3.37)We also have : ( ψ ◦ β )[ x, [ y, z ]] = − h y, [ z, ψ ◦ β ( x )] i + h z, [ y, ψ ◦ β ( x )] i − h x, [ z, ψ ◦ β ( y )] i + h x, [ y, ψ ◦ β ( z )] i (3.38)Now we have : ( ψ ◦ β − ψ ◦ β )[ x, [ y, z ]] = − h y, [ z, ψ ◦ β ( x )] i + h z, [ y, ψ ◦ β ( x )] i − h x, [ z, ψ ◦ β ( y )] i + h x, [ y, ψ ◦ β ( z )] i + h y, [ z, ψ ◦ β ( x )] i − h z, [ y, ψ ◦ β ( x )] i + h x, [ z, ψ ◦ β ( y )] i − h x, [ y, ψ ◦ β ( z )] i = − h y, [ z, ( ψ ◦ β − ψ ◦ β )( x )] i + h z, [ y, ( ψ ◦ β − ψ ◦ β )( x )] i − h x, [ z, ( ψ ◦ β − ψ ◦ β )( y )] i + h x, [ y, ( ψ ◦ β − ψ ◦ β )( z )] i = h ( ψ ◦ β − ψ ◦ β )( x ) , [ y, z ] i + h x, [( ψ ◦ β − ψ ◦ β )( y ) , z ] i + h x, [ y, ( ψ ◦ β − ψ ◦ β )( z )] i (3.39)Then ( ψ ◦ β − ψ ◦ β ) is a prederivation of G .We are now going to verify if ( β ◦ ψ − β ◦ ψ ) satisfies relation (3.20). Let x and y betwo elements of G and f be in G ∗ . [ β ◦ ψ − β ◦ ψ , ad ∗ x ◦ ad ∗ y ]( f ) = [ β ◦ ψ , ad ∗ x ◦ ad ∗ y ]( f ) − [ β ◦ ψ , ad ∗ x ◦ ad ∗ y ]( f )= (cid:16) β ◦ ψ ◦ ad ∗ x ◦ ad ∗ y − ad ∗ x ◦ ad ∗ y ◦ β ◦ ψ (cid:17) ( f ) − (cid:16) β ◦ ψ ◦ ad ∗ x ◦ ad ∗ y − ad ∗ x ◦ ad ∗ y ◦ β ◦ ψ (cid:17) ( f )= (cid:16) β ◦ ad x ◦ ad y ◦ ψ − ad ∗ x ◦ ad ∗ y ◦ β ◦ ψ (cid:17) ( f ) − (cid:16) β ◦ ad x ◦ ad y ◦ ψ − ad ∗ x ◦ ad ∗ y ◦ β ◦ ψ (cid:17) ( f ) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reautomorphisms of T ∗ G = β (cid:16) [ x, [ y, ψ ( f )]] (cid:17) − ad ∗ x ◦ ad ∗ y (cid:16) β (cid:0) ψ ( f ) (cid:1)(cid:17) − β (cid:16) [ x, [ y, ψ ( f )]] (cid:17) + ad ∗ x ◦ ad ∗ y (cid:16) β (cid:0) ψ ( f ) (cid:1)(cid:17) = − ad ∗ [ y,ψ ( f )] (cid:16) β ( x ) (cid:17) − ad ∗ x ◦ ad ∗ ψ ( f ) (cid:16) β ( y ) (cid:17) + ad ∗ x ◦ ad ∗ y (cid:16) β (cid:0) ψ ( f ) (cid:1)(cid:17) − ad ∗ x ◦ ad ∗ y (cid:16) β (cid:0) ψ ( f ) (cid:1)(cid:17) + ad ∗ [ y,ψ ( f )] (cid:16) β ( x ) (cid:17) + ad ∗ x ◦ ad ∗ ψ ( f ) (cid:16) β ( y ) (cid:17) − ad ∗ x ◦ ad ∗ y (cid:16) β (cid:0) ψ ( f ) (cid:1)(cid:17) + ad ∗ x ◦ ad ∗ y (cid:16) β (cid:0) ψ ( f ) (cid:1)(cid:17) = − ad ∗ y ◦ ad ∗ ψ ( f ) (cid:16) β ( x ) (cid:17) + ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) − ad ∗ x ◦ ad ∗ ψ ( f ) (cid:16) β ( y ) (cid:17) + ad ∗ y ◦ ad ∗ ψ ( f ) (cid:16) β ( x ) (cid:17) − ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) + ad ∗ x ◦ ad ∗ ψ ( f ) (cid:16) β ( y ) (cid:17) = − ad ∗ y ◦ ad ∗ ψ ( β ( x )) f + ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) − ad ∗ x ◦ ad ∗ ψ ( β ( y )) f + ad ∗ y ◦ ad ∗ ψ ( β ( x )) f − ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) + ad ∗ x ◦ ad ∗ ψ ( β ( y )) f (3.40)Hence, [ β ◦ ψ − β ◦ ψ , ad ∗ x ◦ ad ∗ y ]( f ) = ad ∗ y ◦ ad ∗ ( ψ ◦ β − ψ ◦ β )( x ) f + ad ∗ x ◦ ad ∗ ( ψ ◦ β − ψ ◦ β )( y ) f + ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) − ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) (3.41)The map ( β ◦ ψ − β ◦ ψ ) does not satisfy the relation (3.20) (with the prederivation ( ψ ◦ β − ψ ◦ β ) ) because of the term ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) − ad ∗ ψ ( f ) ◦ ad ∗ y (cid:16) β ( x ) (cid:17) . ξ : G ∗ → G ∗ Proposition 3.3.1. Let G be a Lie algebra and let α be a prederivation of G . A linear map ξ ′ : G → G satisfies [ ξ ′ , ad x ◦ ad y ] = − (cid:0) ad α ( x ) ◦ ad y + ad x ◦ ad α ( y ) (cid:1) , for any elements x and y of G if and only if there exists a linear map j : G → G satisfying [ j, ad x ◦ ad y ] = 0 , (3.42) for any x , y and z in G , such that ξ ′ = j − α .Proof. Let ξ ′ and α be as in the Proposition 3.3.1. For any x, y, z in G , one has ξ ′ ◦ ad x ◦ ad y ( z ) − ad x ◦ ad y ◦ ξ ′ ( z ) = − ad α ( x ) ◦ ad y ( z ) − ad x ◦ ad α ( y ) ( z )= − (cid:2) α ( x ) , [ y, z ] (cid:3) − (cid:2) x, [ α ( y ) , z ] (cid:3) = − α (cid:0) [ x, [ y, z ]] (cid:1) + [ x, [ y, α ( z )]]= − α ◦ ad x ◦ ad y ( z ) + ad x ◦ ad y ◦ α ( z ) . We then have h ( ξ ′ + α ) , ad x ◦ ad y i = 0 , (3.43)for any elements x, y of G . To finish the proof we just take j = ξ ′ + α . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras Let us note by J ′ the space of linear maps j : G → G which satisfy [ j, ad x ◦ ad y ] = 0 , (3.44)for every x, y, z in G . Proposition 3.3.2. Let G be a non Abelian Lie algebra with dual space G ∗ . A linear map ξ : G ∗ → G ∗ satisfies [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) , for some prederivation α of G and every x, y in G if and only if its transpose ξ t : G → G is of the form ξ t = j − α ,where j is in J ′ .Proof. Consider a linear map ξ satisfying the hypothesis of Proposition 3.3.2. That is, forevery x, y in G , [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) , for some α in Pder ( G ) . Takingtransposes of the two sides one has [ ξ, ad ∗ x ◦ ad ∗ y ] t = (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) (cid:17) t − h ξ t , (cid:0) ad ∗ x ◦ ad ∗ y (cid:1) t i = (cid:16) ad ∗ α ( x ) ◦ ad ∗ y (cid:17) t + (cid:16) ad ∗ x ◦ ad ∗ α ( y ) (cid:17) t − h ξ t , ad y ◦ ad x i = ad y ◦ ad α ( x ) + ad α ( y ) ◦ ad x h ξ t , ad y ◦ ad x i = − (cid:16) ad y ◦ ad α ( x ) + ad α ( y ) ◦ ad x (cid:17) . (3.45)From Proposition 3.3.1 we conclude that ξ t = j − α , where j is in J ′ .As a consequence, we have the following corollary of Theorem 3.3.1. Corollary 3.3.1. A prederivation p : T ∗ G → T ∗ G is defined by : p ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + f ◦ ( j − α ) (cid:17) (3.46) for any element ( x, f ) of T ∗ G , where α : G → G is a prederivation of G , j : G → G is in J ′ , β : G → G ∗ and ψ : G ∗ → G are linear maps satisfying relations (3.17), (3.18) and (3.19). α , β , ψ , ξ in orthogonal Lie algebras All over this section we consider an orthogonal Lie algebra ( G , µ ) . Let θ : G → G ∗ still standfor the isomorphism defined by h θ ( x ) , y i := µ ( x, y ) , for all x, y in G . Lemma 3.4.1. The map β α β := θ − ◦ β is an isomorphism between the space Q ′ oflinear maps β satisfying relation (3.17) and the space Pder ( G ) of prederivations of G .Proof. Let β be an element of Q ′ . For any x, y, z in G , we have α β (cid:0)(cid:2) x, [ y, z ] (cid:3)(cid:1) := θ − ◦ β (cid:0)(cid:2) x, [ y, z ] (cid:3)(cid:1) = θ − (cid:16) − ad ∗ [ y,z ] (cid:0) β ( x ) (cid:1) + ad ∗ x ◦ ad ∗ y (cid:0) β ( z ) (cid:1) − ad ∗ x ◦ ad ∗ z (cid:0) β ( y ) (cid:1)(cid:17) = − ad [ y,z ] ◦ θ − ◦ β ( x )+ ad x ◦ ad y ◦ θ − ◦ β ( z ) − ad x ◦ ad z ◦ θ − ◦ β ( y ) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras = − (cid:2) [ y, z ] , α β ( x ) (cid:3) + (cid:2) x, [ y, α β ( z )] (cid:3) − (cid:2) x, [ z, α β ( z )] (cid:3) (3.47)Then α β belongs to Pder ( G ) . Conversely, let α be a prederivation of G and set β α := θ ◦ α .For any x, y, z in G one has β α (cid:0)(cid:2) x, [ y, z ] (cid:3)(cid:1) := θ (cid:16)(cid:2) α ( x ) , [ y, z ] (cid:3) + (cid:2) x, [ α ( y ) , z ] (cid:3) + (cid:2) x, [ y, α ( z )] (cid:3)(cid:17) = − θ ◦ ad [ x,y ] (cid:0) α ( x ) (cid:1) + θ ◦ ad x (cid:0) [ α ( y ) , z ] (cid:1) + θ ◦ ad x (cid:0) [ y, α ( z )] (cid:1) = − ad ∗ [ y,z ] ( θ ◦ α )( x ) + ad ∗ x ◦ θ (cid:0) − ad z ( α ( y )) (cid:1) + ad ∗ x ◦ θ ◦ ad y (cid:0) α ( z ) (cid:1) = − ad ∗ [ y,z ] β α ( x ) − ad ∗ x ◦ ad ∗ y β α ( y ) + ad ∗ x ◦ ad ∗ y β α ( z ) Then β α is an element of Q ′ . This correspondence is obviously bijective.Now we are going to look at maps ψ ′ s in the case where G is an orthogonal Lie algebra. Lemma 3.4.2. The map ψ j ψ := ψ ◦ θ is an isomorphism between the space of linearmaps ψ : G ∗ → G which satisfy relations (3.18) and the space J ′ .Proof. Take ψ as in the Lemma 3.4.2. Then for any elements x, y, z in G , we have [ ψ ◦ θ, ad x ◦ ad y ] = ψ ◦ θ ◦ ad x ◦ ad y − ad x ◦ ad y ◦ ψ ◦ θ = ψ ◦ ad ∗ x ◦ ad ∗ y ◦ θ − ψ ◦ ad ∗ x ◦ ad ∗ y ◦ θ = 0 Then j ψ := ψ ◦ θ is an element of J ′ .Now consider an element j of J ′ and set ψ j := j ◦ θ − . Taking two arbitrary elements x and y of G , we have ψ j ◦ ad ∗ x ◦ ad ∗ y = j ◦ θ − ◦ ad ∗ x ◦ ad ∗ y = j ◦ ad x ◦ ad y ◦ θ − = ad x ◦ ad y ◦ j ◦ ψ = ad x ◦ ad y ◦ ψ j (3.48)Then ψ j satisfies relation (3.18). This correspondence is linear and invertible. Lemma 3.4.3. The maps ξ θ − ◦ ξ ◦ θ is an isomorphism of Lie algebras between the Liealgebra E ′ of linear maps ξ : G ∗ → G ∗ satisfying [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) ,for some prederivation α of G and any x and y in G , and the Lie algebra S ′ of linear maps ξ ′ : G → G such that [ ξ ′ , ad x ◦ ad y ] = ad α ( x ) ◦ ad y + ad x ◦ ad α ( y ) , for some prederivation α of G and every elements x, y of G .Proof. Consider an element ξ of E ′ and set δ ξ := θ − ◦ ξ ◦ θ . For x, y, z in G we have [ δ ξ , ad x ◦ ad y ] = δ ξ ◦ ad x ◦ ad y − ad x ◦ ad y ◦ δ ξ = θ − ◦ ξ ◦ θ ◦ ad x ◦ ad y − ad x ◦ ad y ◦ θ − ◦ ξ ◦ θ = θ − ◦ ξ ◦ ad ∗ x ◦ ad ∗ y ◦ θ − θ − ◦ ad ∗ x ◦ ad ∗ y ◦ ξ ◦ θ = θ − ◦ ( ξ ◦ ad ∗ x ◦ ad ∗ y − ad ∗ x ◦ ad ∗ y ◦ ξ ) ◦ θ = θ − ◦ [ ξ, ad ∗ x ◦ ad ∗ y ] ◦ θ = θ − ◦ (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) (cid:17) ◦ θ, for some α ∈ Pder ( G ) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras = (cid:16) ad α ( x ) ◦ ad y + ad x ◦ ad α ( y ) (cid:17) ◦ θ − ◦ θ = ad α ( x ) ◦ ad y + ad x ◦ ad α ( y ) . (3.49)Then δ ξ is an element of S ′ .Now consider an element ξ ′ of S ′ with α as corresponding prederivation of G . Then ξ := θ ◦ ξ ′ ◦ θ − is an element of E ′ as one can see in the following. For any elements x, y in G we have [ ξ, ad ∗ x ◦ ad ∗ y ] = ξ ◦ ad ∗ x ◦ ad ∗ y − ad ∗ x ◦ ad ∗ y ◦ ξ = θ ◦ ξ ′ ◦ θ − ◦ ad ∗ x ◦ ad ∗ y − ad ∗ x ◦ ad ∗ y ◦ θ ◦ ξ ′ ◦ θ − = θ ◦ ξ ′ ◦ ad x ◦ ad y ◦ θ − − θ ◦ ad x ◦ ad y ◦ ξ ′ ◦ θ − = θ ◦ (cid:0) ξ ′ ◦ ad x ◦ ad y − ad x ◦ ad y ◦ ξ ′ (cid:1) ◦ θ − = θ ◦ [ ξ ′ , ad x ◦ ad y ] ◦ θ − = θ ◦ (cid:0) ad α ( x ) ◦ ad y + ad x ◦ ad α ( y ) (cid:1) ◦ θ − = (cid:16) ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) (cid:17) ◦ θ ◦ θ − = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) . It comes that ξ := θ ◦ ξ ′ ◦ θ − belongs to E ′ .Now note by δ : E ′ → S ′ , ξ δ ξ := θ − ◦ ξ ◦ θ . For any ξ and ξ in E ′ , we have : [ δ ξ , δ ξ ] = δ ξ ◦ δ ξ − δ ξ ◦ δ ξ = θ − ◦ ξ ◦ θ ◦ θ − ◦ ξ ◦ θ − θ − ◦ ξ ◦ θ ◦ θ − ◦ ξ ◦ θ = θ − ◦ ξ ◦ ξ ◦ θ − θ − ◦ ξ ◦ ξ ◦ θ = θ − ◦ [ ξ , ξ ] ◦ θ = δ [ ξ ,ξ ] Thus, the map ξ θ − ◦ ξ ◦ θ is an isomorphism of Lie algebras.Now we have the following result which states that Pder ( T ∗ G ) is completely determinedby Pder ( G ) and J ′ . Proposition 3.4.1. Let ( G , µ ) be an orthogonal Lie algebra and G ∗ its dual space. Considerthe isomorphism θ : G → G ∗ defined by h θ ( x ) , y i := µ ( x, y ) . Any prederivation φ of T ∗ G has the following form φ ( x, f ) = (cid:16) α ( x ) + j ◦ θ − ( f ) , θ ◦ α ( x ) + ( j t − α t )( f ) (cid:17) , (3.50) for every ( x, f ) in T ∗ G , where- α , α are prederivations of G ,- j , j are in J ′ , with ad ∗ x (cid:0) ad ∗ j ◦ θ − ( f ) g − ad ∗ j ◦ θ − ( g ) f (cid:1) = 0 , for all x in G ; f, g in G ∗ ,- j t and α t are the transposes of j and α respectively. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras 1. Recall relation (3.18) : ψ ◦ ad ∗ x ◦ ad ∗ y = ad x ◦ ad y ◦ ψ, for all x and y in G . We can also write ψ ◦ ad ∗ y ◦ ad ∗ x = ad y ◦ ad x ◦ ψ, for all x and y in G . Substracting the above two relations we have, for all elements x, y of G , ψ ◦ [ ad ∗ x , ad ∗ y ] = [ ad x , ad y ] ◦ ψ which can be written ψ ◦ ad ∗ [ x,y ] = ad [ x,y ] ◦ ψ. Hence, if ψ satisfies relation (3.18) then ψ ◦ ad ∗ [ x,y ] = ad [ x,y ] ◦ ψ, (3.51) for any x and y in G .2. By the same way, we recall relation (3.20) : [ ξ, ad ∗ x ◦ ad ∗ y ] = ad ∗ α ( x ) ◦ ad ∗ y + ad ∗ x ◦ ad ∗ α ( y ) for any x and y in G . Changing the roles played by x and y we obtain [ ξ, ad ∗ y ◦ ad ∗ x ] = ad ∗ α ( y ) ◦ ad ∗ x + ad ∗ y ◦ ad ∗ α ( x ) . Substracting again the two last relations above, we have h ξ, [ ad ∗ x , ad ∗ y ] i = (cid:2) ad ∗ α ( x ) , ad ∗ y (cid:3) + (cid:2) ad ∗ x , ad ∗ α ( y ) (cid:3)h ξ, ad ∗ [ x,y ] i = ad ∗ [ α ( x ) ,y ] + ad ∗ [ x,α ( y )] . That is h ξ, ad ∗ [ x,y ] i = ad ∗ ([ α ( x ) ,y ]+[ x,α ( y )]) , (3.52) for all x and y in G . A semi-simple Lie algebra is an orthogonal Lie algebra. Moreover, if G is semi-simple, thenany prederivation is a derivation and hence an inner derivation ([64]). In this case, we canwrite relation (3.52) as follows h ξ, ad ∗ [ x,y ] i = ad ∗ α ([ x,y ]) (3.53)for all x, y in G . Since [ G , G ] = G , we can simply write [ ξ, ad ∗ x ] = ad ∗ α ( x ) (3.54)for any x of G . The following lemma is an immediate consequence of Section 2.4. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras If G is a semi-simple Lie algebra, then E ′ is nothing but the space of linearmaps ξ ′ : G ∗ → G ∗ such that [ ξ, ad ∗ x ] = ad ∗ α ( x ) , for some derivation α of G and any x in G . Furthermore, if G decomposes into simple idealas follows G = s ⊕ s ⊕ · · · ⊕ s p ( p ∈ N ), then ξ = ad ∗ x + p M i =1 λ i id s ∗ i , (3.55) for some x in G and some real numbers λ , λ , . . . , λ p . Now we work on the maps β and the relation (3.17). The maps β α β := θ − ◦ β defined in Lemma 3.4.1 is then an isomorphism between the spaces der ( G ) and Q ′ , sincePder ( G ) = der ( G ) . From Proposition 2.4.3 it comes that if α β is a derivation then β = θ ◦ α β is a -cocycle of G with values in G ∗ for the coadjoint representation of G on G ∗ . We thenproved the following lemma. Lemma 3.4.5. If G is a semi-simple Lie algebra, then any element β of Q ′ is a -cocycleof G with values in G ∗ for the coadjoint representation of G on G ∗ . Lemma 3.4.6. If G is a semi-simple Lie algebra, then Ψ ′ = { } .Proof. Relation (3.19) means that the linear form ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f is closed, for any linearforms f and g on G . Since G is semi-simple, it is perfect ; that is G is equal to its derivedideal ( [ G , G ] = G ). It is known that any closed form on a perfect Lie algebra is zero. Then,the closed form ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f is equal to zero for any f and g in G ∗ , i.e. ad ∗ ψ ( f ) g − ad ∗ ψ ( g ) f = 0 , (3.56)for every f and g in G . Again because [ G , G ] = G , Relation (3.18) becomes ψ ◦ ad ∗ x = ad x ◦ ψ (3.57)Now we conclude with Proposition 2.4.4.We summarize all the above by the following Theorem 3.4.1. Let G be a finite dimensional semi-simple Lie group with Lie algebra G .Then every prederivation of the Lie algebra T ∗ G of the cotangent bundle Lie group T ∗ G of G is a derivation.Proof. This is direct consequence of Lemmas 3.4.4, 3.4.5, 3.4.6 and the fact that anyprederivation of a semi-simple Lie algebra is a derivation. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras Lemma 3.4.7. Let G be a compact Lie algebra. Then every prederivation α of G is of theform α = ad x ⊕ ϕ , where x belongs to the derived ideal of [ G , G ] and ϕ is an endomorphismof Z ( G ) . That is for any x in [ G , G ] and x in Z ( G ) , α ( x + x ) = [ x , x ] + ϕ ( x ) , (3.58) where x in [ G , G ] , where ϕ is an endomorphism of Z ( G ) .Proof. Recall that a compact Lie algebra is a Lie algebra G which decomposes into the sum G = [ G , G ] ⊕ Z ( G ) of its derived ideal and its centre, with [ G , G ] semi-simple and compact.A prederivation α of such a Lie algebra preserves each of the factor [ G , G ] and Z ( G ) . Thenit can be written as a direct sum of a prederivation α of [ G , G ] and a prederivation α of Z ( G ) . Since [ G , G ] is semi-simple, the prederivation α is an inner derivation. Furthermore,the derivation α of Z ( G ) is just an endomorphism of Z ( G ) because Z ( G ) is an Abelianideal of G . Hence, we can write α = ad x ⊕ ϕ, (3.59)where x is an element of the derived ideal of G and ϕ is an endomorphism of Z ( G ) . Theprederivation α acts on G as follows. If x = x + x with x in [ G , G ] and x in Z ( G ) , α ( x ) = α ( x + x ) = [ x , x ] + ϕ ( x ) . (3.60) Lemma 3.4.8. If G is a compact Lie algebra. Then the space Ψ ′ is isomorphic to the space End ( Z ( G )) of endomorphisms of the centre Z ( G ) of G .Proof. A compact Lie algebra admits an orthogonal structure, then from Lemma 3.4.2the space of linear maps ψ : G ∗ → G which are equivariant with respect to the adjointand coadjoint representation of [ G , G ] on G and G ∗ respectively is isomorphic to the space J ′ . The correspondence is given by ψ = j ◦ θ − , for some j in J ′ . For any element x of G = [ G , G ] ⊕ Z ( G ) , we write x = x + x , where x is in the semi-simple ideal [ G , G ] while x belongs to the centre Z ( G ) of G . Set j ( x ) = (cid:0) j ( x ) + j ( x ) (cid:1)| {z } ∈ [ G , G ] + (cid:0) j ( x ) + j ( x ) (cid:1)| {z } ∈ Z ( G ) , (3.61)where j : [ G , G ] → [ G , G ] , j : [ G , G ] → Z ( G ) , j : Z ( G ) → [ G , G ] and j : Z ( G ) → Z ( G ) are linear maps. For any x, y, z in G , we have j (cid:16)(cid:2) [ x, y ] , z (cid:3)(cid:17) = (cid:2) [ x, y ] , j ( z ) (cid:3) j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17) = (cid:2) [ x , y ] , j ( z ) + j ( z ) (cid:3) = (cid:2) [ x , y ] , j ( z ) (cid:3) + (cid:2) [ x , y ] , j ( z ) (cid:3) (3.62)If we take z = 0 , then j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17) = (cid:2) [ x , y ] , j ( z ) (cid:3) for all x , y , z in [ G , G ] . Relation(3.62) gives (cid:2) [ x , y ] , j ( z ) (cid:3) = 0 for all x , y in [ G , G ] and z in Z ( G ) ; that is j ( z ) belongs On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras to the centre of [ G , G ] which is { } since [ G , G ] is semi-simple. It comes that j ≡ . Onthe other way j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17) = j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17) + j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17)(cid:2) [ x , y ] , j ( z ) (cid:3)| {z } ∈ [ G , G ] = j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17)| {z } ∈ [ G , G ] + j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17)| {z } ∈ Z ( G ) (3.63)It comes that j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17) = 0 , for all x , y , z in [ G , G ] . Then J ≡ on the semi-simple ideal [ G , G ] . Now we have just j ( x ) = j ( x ) + j ( x ) , (3.64)where j is an endomorphism of [ G , G ] satisfying j (cid:16)(cid:2) [ x , y ] , z (cid:3)(cid:17) = (cid:2) [ x , y ] , j ( z ) (cid:3) , (3.65)for all x , y , z in [ G , G ] ; and j is in End ( Z ( G )) . Since [ G , G ] is perfect then (3.65) canbe written j ([ x , y ]) = [ x , j ( y )] , (3.66)for all x , y in [ G , G ] . It comes, from Corollary 2.4.4, that j ( x ) = p X i =1 λ i x i + j ( x ) , (3.67)where x = x + x + · · · + x p is the decomposition of x into elements of the simplecomponents of [ G , G ] . Now we have, ψ ( f ) = j ◦ θ − ( f )= j (cid:16) p X k =1 θ − ( f k ) + θ − ( f ) (cid:17) = p X i =1 λ i θ − ( f i ) + j ◦ θ − ( f ) (3.68)From Lemma 3.4.6 the restriction of ψ to the semi-simple ideal must be zero, then ψ ( f ) = j ◦ θ − ( f ) . (3.69)and we are done. Lemma 3.4.9. If G is a compact Lie algebra, then the space Q ′ is isomorphic to the space ad [ G , G ] ⊕ End ( Z ( G )) , where ad [ G , G ] stands for the space of inner derivations of the derivedideal [ G , G ] of G .Proof. The proof is straightforward. Lemma 3.4.1 asserts that Q ′ is isomorphic to Pder ( G ) and Lemma 3.4.7 implies that Pder ( G ) ∼ = ad [ G , G ] ⊕ End ( Z ( G )) . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras Let G be a compact Lie algebra. Then any linear map ξ : G ∗ → G ∗ satisfying relation (3.20) can be written as ξ = p M i =1 λ i id ∗ s i + ad ∗ x ! ⊕ η, (3.70) where η is an endomorphism of Z ( G ∗ ) , x is an element of [ G , G ] = s ⊕ s · · · ⊕ s p ; λ , λ , . . . , λ p are real numbers and p is the number or simple components of [ G , G ] . Moreprecisely, if f = f + f is an element of G ∗ with f = f + f + f + · · · + f p in [ G , G ] ∗ = s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p and f in Z ( G ) ∗ , then ξ ( f ) = p X i =1 λ i f i + ad ∗ x f + η ( f ) . (3.71) Proof. We have already seen that the transpose ξ t of ξ has the form ξ t = j − α , where j isin J ′ . From the proof of Lemma 3.4.8 (see Relation (3.67)) we have j = p M i =1 λ i id s i ! ⊕ ρ , where ρ is an endomorphism of Z ( G ) . Hence, ξ t = p M i =1 λ i id s i ! ⊕ ρ − (cid:0) ad x ⊕ ϕ (cid:1) , since a prederivation α of the compact Lie algebra G is given by α = ad x ⊕ ϕ , where x is in [ G , G ] and ϕ is an endomorphism of Z ( G ) (see Lemma 3.4.7). We simply write ξ t = p M i =1 λ i id s i − ad x ! ⊕ ϕ , where ϕ = ρ + ϕ belongs to End ( Z ( G )) . It comes that ξ = p M i =1 λ i id s ∗ i + ad ∗ x ! ⊕ η, where η = ϕ t is the transpose of ϕ .The following Proposition holds and is a direct consequence of the Lemmas above. Proposition 3.4.2. Let G be a compact Lie algebra. Let θ : G → G ∗ be the isomorphismdefined by h θ ( x ) , y i := µ ( x, y ) , where µ is an arbitrary orthogonal structure on G . Then anyprederivation of T ∗ G has the following form : φ ( x, f ) = (cid:16) [ x , x ]+ ϕ ( x )+ ϕ ◦ θ − ( f ) , θ ([ y , x ])+ θ ◦ ϕ ( x )+ p X i =1 λ i f i + ad ∗ x f + η ( f ) (cid:17) , (3.72) where On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 rthogonal Lie algebras - x , y are in [ G , G ] ;- ϕ , ϕ , ϕ are endomorphisms of Z ( G ) ;- η is in End (cid:0) Z ( G ) ∗ (cid:1) ;- ν i , λ i , i = 1 , , . . . , p are real numbers;- x = x + x = x + x + · · · + x p + x ∈ G = [ G , G ] ⊕ Z ( G ) - and f = f + f = f + f + · · · + f p + f ∈ G ∗ = [ G , G ] ∗ ⊕ Z ( G ) ∗ ; p being the number of simple components of [ G , G ] = s ⊕ s ⊕ · · · ⊕ s p .Proof. Let G = [ G , G ] ⊕ Z ( G ) be a compact Lie algebra. From Proposition 3.4.1 we havethat any prederivation φ of G has the form φ ( x, f ) = (cid:16) α ( x ) + j ◦ θ − ( f ) , θ ◦ α ( x ) + ( j t − α t )( f ) (cid:17) , for every ( x, f ) in T ∗ G , where α , α are prederivations of G , j , j are in J ′ , j t and α t are thetransposes of j and α respectively. Now Lemma 3.4.7 implies that α ( x ) = [ x , x ]+ ϕ ( x ) and α ( x ) = [ y , x ] + ϕ ( x ) , for every x = x + x ∈ G = [ G , G ] ⊕ Z ( G ) where x and y are fix elements of the derived ideal [ G , G ] . From the proof of Lemma 3.4.8, we have j ( x ) = p X i =1 ν i x i + ϕ ( x ) ,j ( x ) = p X i =1 λ i x i + ϕ ′ ( x ) , where x = x + x = x + x + · · · + x p + x ∈ [ G , G ] ⊕ Z ( G ) , λ, ν i , i = 1 , , . . . , p beingreal numbers. Then, for any f = f + f = f + f + · · · + f p + f in [ G , G ] ∗ ⊕ Z ( G ) ∗ , wehave j ◦ θ − ( f ) = j (cid:0) θ − ( f ) (cid:1) = p X i =1 ν i (cid:16) θ − ( f i ) (cid:17) + ϕ (cid:0) θ − ( f ) (cid:1) . (3.73)The restriction of ψ := j ◦ θ − to the dual space of the semi-simple ideal [ G , G ] must bezero (see Proposition 2.4.4). Then we have j ◦ θ − ( f ) = ϕ ◦ θ − ( f ) , (3.74)for all f = f + f = f + f + · · · + f p + f in [ G , G ] ∗ ⊕ Z ( G ) ∗ .Let us, now have a look at maps β := θ ◦ α . We have β ( x ) = θ (cid:0) [ y , x ] (cid:1) + θ ◦ ϕ ( x ) (3.75)for all x = x + x in G = [ G , G ] ⊕ Z ( G ) .We finish this chapter by giving some situations where prederivations are used. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ossible Applications and Examples Example 3.5.1 (Affine Lie algebra of the real line) . Let Aff ( R ) stand for the affine Liegroup of the real line and note by aff ( R ) its Lie algebra. We recall (see Example 2.5.1) that D := T ∗ aff ( R ) = span ( e , e , e ) with the following brackets [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , One can readily verify that Pder ( D ) = der ( D ) = R ⋉ R (semi-direct product of theAbelian Lie algebras R = span R ( φ , φ , φ ) and R = span R ( φ , φ ) ) with the followingbrackets (see Example 2.5.1) : [ φ , φ ] = φ , [ φ , φ ] = φ , [ φ , φ ] = φ , [ φ , φ ] = φ . Example 3.5.2 (The Lie Algebra of the Group SL (2 , R ) of Special Linear Group andthe Lie Algebra of the Group SO (3) of Rotations) . The Lie algebra sl (2 , R ) is simple,then Pder ( T ∗ sl (2 , R )) = der ( T ∗ sl (2 , R )) (See Example 2.5.3). By the same argument,Pder ( T ∗ so (3 , R )) = der ( T ∗ so (3 , R )) (see Example 2.5.2). Example 3.5.3 (The -dimensional Oscillator Group) . In Example 1.1.3 we have definethe Oscillator Lie group and its Lie algebra. The -dimensional oscillator algebra, is thespace G λ = span { e − , e , e , ˇ e } ( λ > ) with the following brackets : [ e − , e ] = λ ˇ e ; [ e − , ˇ e ] = − λe ; [ e , ˇ e ] = e . (3.76)Let ( e ∗− , e ∗ , e ∗ , ˇ e ∗ ) stand for the basis of G ∗ λ dual to ( e − , e , e , ˇ e ) . Then, the Lie algebra T ∗ G λ = span ( e − , e , e , ˇ e , e ∗− , e ∗ , e ∗ , ˇ e ∗ ) with the brackets [ e − , e ] = λ ˇ e ; [ e − , ˇ e ] = − λe ; [ e , ˇ e ] = e [ e − , e ∗ ] = λ ˇ e ∗ ; [ e − , ˇ e ∗ ] = − λe ∗ ; [ e , e ∗ ] = − ˇ e ∗ [ˇ e , e ∗ ] = − λe ∗− ; [ e , ˇ e ∗ ] = λe ∗− ; [ˇ e , e ∗ ] = e ∗ (3.77)Consider the form µ λ defined on G λ by µ λ ( x, y ) = x − y + x y − + 1 λ ( x y + ˇ x ˇ y ) (3.78)for all x = x − e − + x e + x e + ˇ x ˇ e and y = y − e − + y e + y e + ˇ y ˇ e . It is readilycheiked that the form (3.78) defines an orthogonal structure on G λ ([12]). The isomorphism θ : G λ → G λ defined by h θ ( x ) , y i = µ λ ( x, y ) , for all x, y in G λ is given by θ ( e − ) = e ∗ , θ ( e ) = e ∗− , θ ( e ) = 1 λ e ∗ , θ (ˇ e ) = 1 λ ˇ e ∗ , (3.79)The inverse map of θ reads : θ − ( e ∗− ) = e , θ − ( e ∗ ) = e − , θ − ( e ∗ ) = λe , θ − (ˇ e ∗ ) = λ ˇ e . (3.80) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ossible Applications and Examples Since G λ is an orthogonal Lie algebra, any prederivation φ of T ∗ G λ can be written asfollows. φ ( x, f ) = (cid:16) α ( x ) + j ◦ θ − ( f ) , θ ◦ α ( x ) + ( j t − α t )( f ) (cid:17) , for every ( x, f ) in T ∗ G , where α , α are in Pder ( G λ ) and j , j are in J ′ with conditionslisted in Proposition 3.4.1. Now we have : • A prederivation α of G λ can be represented in the basis ( e − , e , e , ˇ e ) by the followingmatrix. α = a a a a − λa a a − λa − a a , (3.81)where a ij ’s are reals numbers. We can put α = a a a a − λa a a − λa − a a ; α = b b b b − λb b b − λb − b b . • A linear map j : G λ → G λ which satisfies (3.44) has the following form : j = j j j j 00 0 0 j , (3.82)where j and j are real numbers. We set j = a a a a 00 0 0 a ; j = b b b b 00 0 0 b . It follows that j ◦ θ − = a a a λa 00 0 0 λa ; θ ◦ α = b b b b − b λ b 33 1 λ b − b − λ b 34 1 λ b . If we consider the fact that ad ∗ x (cid:0) ad ∗ j ◦ θ − ( f ) g − ad j ◦ θ − ( g ) f (cid:1) = 0 , for all x in G λ and any f, g in G ∗ λ , we obtain j ◦ θ − = a On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ossible Applications and Examples Now we write the matrix of a prederivation φ of T ∗ G λ . φ = a a a a a − λa a a − λa − a a b b b b b b − a λa λa b − a − b λ b 33 1 λ b − a b − a a − b − λ b 34 1 λ b − a − a b − a (3.83)It comes that Pder ( T ∗ G λ ) = span { φ i , ≤ i ≤ } , where φ = e − e φ = 2 e + e + e − e − e − e φ = − e + λe − λe + e φ = e − λe + λe − e φ = e φ = − λe + λe − λe + λe φ = e φ = 2 e + λ e + λ e φ = − λe + λe φ = λe − λe φ = e + e + e + e φ = e φ = − e + e (3.84)with the following brackets [ φ , φ ] = − φ [ φ , φ ] = − φ [ φ , φ ] = φ [ φ , φ ] = φ [ φ , φ ] = 2 φ [ φ , φ ] = − φ [ φ , φ ] = − φ [ φ , φ ] = − φ [ φ , φ ] = 2 φ [ φ , φ ] = − φ [ φ , φ ] = λφ [ φ , φ ] = − λ φ [ φ , φ ] = φ [ φ , φ ] = − λφ [ φ , φ ] = − λ φ [ φ , φ ] = − φ [ φ , φ ] = − φ [ φ , φ ] = φ [ φ , φ ] = − λφ [ φ , φ ] = − λφ [ φ , φ ] = − φ [ φ , φ ] = − φ [ φ , φ ] = − φ [ φ , φ ] = − φ [ φ , φ ] = φ [ φ , φ ] = φ (3.85)One realizes thatPder ( T ∗ G λ ) = ad T ∗ G λ ⋉ h R ⋉ ( R ⋉ R ) i = ( R ⋉ R ) ⋉ h R ⋉ ( R ⋉ R ) i = (cid:16) span ( φ , φ ) ⋉ span ( φ , φ , φ , φ ) (cid:17) ⋉ h span ( φ , φ ) ⋉ (cid:0) span ( φ , φ ) ⋉ span ( φ , φ , φ (cid:1)i (3.86) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ossible Applications and Examples The existence of affine structures is a difficult and interesting problem ([39],[63], [54], [52]).Let G be a Lie algebra. If D is an invertible derivation of G , one defines an affinestructure by the formula ∇ x y = D − ◦ ad x ◦ D ( y ) , (3.87)for every x and y in G . Unfortunately, there are Lie algebras that admits only singularderivations. Nilpotent such algebras are called characteristically nilpotent Lie algebras.If the Lie algebra G admits no regular derivation, then one uses a regular prederivationwhenever it exists. The idea of the construction is the following ([15]). Given an invertibleprederivation P , set ω ( x, y ) = P ([ x, y ]) − [ x, P ( y )] − [ P ( x ) , y ] , for all x, y ∈ G . Now wedefine ∇ x y = P − ◦ ad x ◦ P ( y ) + 12 P − ◦ ω ( x, y ) , (3.88)for every x, y ∈ G . In general the map x P − ◦ ad x ◦ + P − ◦ ω ( x, · ) might not be arepresentation. However, in many cases, (3.88) gives rise to an affine structure. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 hapter Four Skew-symmetric Prederivations andBi-invariant Metrics of CotangentBundles of Lie Groups Contents T ∗ G . . . . . . . . . . . . . . . . . . . . 794.4 Case of Orthogonal Lie Algebras . . . . . . . . . . . . . . . . . 854.5 Case of Semi-simple Lie Algebras . . . . . . . . . . . . . . . . . 894.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A Lie group endowed with a pseudo-Riemannian metric which is invariant under both leftand right translations, is called an orthogonal, a bi-invariant or a quadratic Lie group. Thecorresponding Lie algebra is called an orthogonal or quadratic Lie algebra.Such Lie groups are important in Mathematics and in Physics. These objects are, forinstance, useful in pseudo-Riemannian geometry, in the theory of Poisson-Lie groups, inrelativity, in the theory of Hamiltonian systems,... ([4], [5], [10], [29], [32], [35], [50]).According to the works of Medina and Revoy ([60],[58]), any orthogonal Lie algebra isobtained by the so-called double extension procedure. This is the case of orthogonal Liealgebras called hyperbolic and whose quadratic form is of signature ( n, n ) (where n is thedimension of the concerned Lie algebra). One particularly interesting case is the one wherethe Lie algebra admits two totally isotropic subalgebras which are in duality. These latterLie algebras named Manin-Lie algebras describe simply connected Poisson-Lie groups.It is known that the cotangent bundle T ∗ G of a Lie group G with Lie algebra G ,considered with its Lie group structure obtained by semi-direct product of the Lie group On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ntroduction G and the Abelian Lie group G ∗ (dual of G ) by means of the co-adjoint representation,possesses a hyperbolic metric : the duality pairing.Here we seek to characterize, up to isometric automorphisms, all orthogonal structureson T ∗ G by means of the duality pairing and adjoint-invariant endomorphisms; hence, all bi-invariant metrics on T ∗ G ; and to determine the corresponding group of isometries. We willfocus our study on orthogonal Lie algebras and on the more particular class of semi-simpleLie algebras.We will take G to be the Lie algebra of a Lie group G , G ∗ will be the dual space of G and T ∗ G := G ⋉ G ∗ will be the semi-direct sum of the Lie algebra G and the vector space G ∗ via the coadjoint representation. Recall that Pder ( G ) stands for the Lie algebra of allprederivations of the Lie algebra G while J ′ is the set of all endomorphisms j of G whichsatisfy j ( (cid:2) [ x, y ] , z (cid:3) ) = (cid:2) [ x, y ] , j ( z ) (cid:3) , for any x, y, z in G .Among others, here are some of the important results contained in this chapter. Theorem A 1. Let G be a Lie algebra. Any orthogonal structure µ on T ∗ G is given by µ (cid:16) ( x, f ) , ( y, g ) (cid:17) = (cid:10) g, j ( x ) (cid:11) + (cid:10) f, j ( y ) (cid:11) + (cid:10) g, j ( f ) (cid:11) + (cid:10) j ( x ) , y (cid:11) , (4.1) where j : G → G , j : G → G ∗ , j : G ∗ → G are as in Proposition 4.3.1.2. If ( G , µ ) is an orthogonal Lie algebra, then any orthogonal structure µ D on T ∗ G hasthe form µ D (cid:16) ( x, f ) , ( y, g ) (cid:17) = (cid:10) g , j ( x ) (cid:11) + (cid:10) f , j ( y ) (cid:11) + (cid:10) g, j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) , y (cid:11) , (4.2) for all ( x, f ) , ( y, g ) in T ∗ G , where θ is the isomorphism induced by the µ through theformula (1.2) and j , j , j satisfy conditions listed in Lemma 4.4.1.3. Let G be a semi-simple Lie algebra. Any orthogonal structure µ on T ∗ G is given by µ (cid:0) ( x, f ) , ( y, g ) (cid:1) = p X i =1 λ i (cid:10) ( x i , f i ) , ( y i , g i ) (cid:11) s i + p X k =1 ν k K k ( x k , y k ) , (4.3) for all x, y ∈ G = s ⊕ s ⊕ · · · ⊕ s p and f, g ∈ G ∗ = s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p ; where λ i ∈ R ∗ , ν i ∈ R and K i stands for the Killing form on s i for all i = 1 , , . . . , p . Theorem B 1. Let G be a Lie algebra and let µ D be an orthogonal structure on T ∗ G . A µ D -skew-symmetric prederivation φ of T ∗ G has the form φ ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ( j ′ t − α t )( f ) (cid:17) , (4.4) where j ′ : G → G is in J ′ , α : G → G is in Pder ( G ) , β : G → G ∗ and ψ : G ∗ → G areas in Theorem 3.3.1, with the additional conditions listed in Proposition 4.3.2. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reliminaries 2. Let ( G , µ ) be an orthogonal Lie algebra. Then any prederivation φ of T ∗ G which isskew-symmetric with respect to any orthogonal structure µ D on T ∗ G can be writtenas follows φ ( x, f ) = (cid:16) α ( x ) + j ′ ◦ θ − ( f ) , θ ◦ α ( x ) + ( j ′ t − α t )( f ) (cid:17) , (4.5) where α , α are in Pder ( G ) ; j ′ , j ′ are in J ′ with the additional conditions listed inProposition 4.4.1.3. Let G a semi-simple Lie algebra. Then any prederivation of T ∗ G which is skew-symmetric with respect to any orthogonal structure µ D on T ∗ G is an inner derivationof T ∗ G . In Section 4.2 is given some basic notions useful to make the chapter understandable.Section 4.3 is dedicated to the characterization of orthogonal structures on the Lie algebrasof cotangent bundles of Lie groups and their groups of isometries. In Sections 4.4 and 4.5are respectively studied the case of orthogonal Lie algebras and the case of semi-simple Liealgebras. The chapter finishes with some examples given in Section 4.6. Let ( G, µ ) be an orthogonal Lie group with Lie algebra G . The metric µ induces on G anadjoint-invariant symmetric non-degenerate bilinear form h , i , i.e. a symmetric and non-degenerate form h , i such that h [ x, y ] , z i + h y, [ x, z ] i = 0 , (4.6)for all x, y in G . The form h , i is called an orthogonal structure on G . It is well known ([60])that any other non-degenerate symmetric bilinear form B on G can be written as B ( x, y ) = h j ( x ) , y i , (4.7)for all x, y in G , where j is a h , i -symmetric automorphism of the vector space G . Further-more, the bilinear form B is adjoint-invariant, i.e. is an orthogonal structure, if and onlyif j commutes with all the adjoint operators ad x , ( x ∈ G ); that is j ([ x, y ]) = [ j ( x ) , y ] = [ x, j ( y )] , (4.8)for every x, y in G .An endomorphism j of G which satisfies Relation (4.8) is called a bi-invariant or anadjoint-invariant endomorphism.From what is said above it comes that if one orthogonal structure h , i is known on G then we will be able to characterize all the orthogonal structures on G by characterizingall the h , i -symmetric bi-invariant tensors on G . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reliminaries Let us begin by some reminders. Definition 4.2.1. Let M be a smooth manifold equipped with a pseudo-Riemannian metric µ . An isometry of the pseudo-Riemannian manifold ( M, µ ) is a diffeomorphism f : M → M such that f ∗ µ = µ , where f ∗ stands for the "pull-back" via f . More precisely, for all x in M and any vectors X | x , Y | x in the tangent space T x M , µ f ( x ) (cid:16) T x f · X | x , T x f · Y | x (cid:17) = µ x (cid:0) X | x , Y | x (cid:1) . (4.9)We denote by I ( M, µ ) the set of all isometries of ( M, µ ) . Nomizu shows in [66] thatthe set Aff ( M, ∇ ) of all affine transformations of the induced affine connection on M ,with the compact-open topology, is a Lie transformation group. But I ( M, µ ) is a closedsubgroup of Aff ( M, ∇ ) . Then, equipped with the compact-open topology, I ( M, µ ) is a Lietransformation group.Let ( G, µ ) be an orthogonal Lie group with Lie algebra G . We note by I ( G, µ ) = I ( G ) the set of all isometries of ( G, µ ) , by F ( G ) the set of those isometries which fix the identityelement ǫ of G and by L G the set of all left translations of G . We recall the following resultdue to Müller. Lemma 4.2.1. ([64]) Let G be a connected Lie group endowed with a bi-invariant metric.1. F ( G ) is a closed Lie subgroup of I ( G ) ;2. L G is a closed connected subgroup isomorphic to G ;3. I ( G ) = L G F ( G ) , with L G ∩ F ( G ) = { id } ; id : G → G is the identity map of G ;4. the manifolds I ( G ) and G × F ( G ) are diffeomorphic. Let ( G, µ ) be an orthogonal Lie group with unit element ǫ and let G be its Lie algebra. Wewill often note by h , i := µ | ǫ the resulting non-degenerate adjoint-invariant bilinear formon G . The fact that G is orthogonal is equivalent to the one that the geodesics through ǫ are the one-parameter subgroups of G (see [38, Exercise 5, page ]).Let exp : G → G denote the exponential map of the Lie group G . Consider an openneighborhood U of in G such that the restriction exp | U : U → exp( U ) of exp to U is adiffeomorphism and note by log : exp( U ) → U the inverse of exp | U . Definition 4.2.2. A local isometry at ǫ is a diffeomorphism ϕ : V → V between two openneighborhoods V and V of ǫ in G such that • ϕ fixes ǫ , i.e. ϕ ( ǫ ) = ǫ ; • ϕ ∗ µ = µ on V , where ϕ ∗ stands for the pull-back via ϕ . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 reliminaries Now, let ϕ be a local isometry at ǫ and x any arbitrary element of G . The local isometry ϕ maps any geodesic through ǫ onto a geodesic through ǫ on a neighborhood of ǫ . Thenthere exists an element y of G such that ϕ (cid:0) exp( tx ) (cid:1) = exp( ty ) , (4.10)for t small enough. Derivating relation (4.10) with respect to t at t = 0 , one has T ǫ ϕ · x = y, (4.11)where T ǫ ϕ is the tangent linear map of ϕ at ǫ . From relation (4.10) we have y = log ◦ ϕ ◦ exp( x ) . (4.12)Relations (4.10) and (4.11) give the following nice formula ϕ = exp ◦ T ǫ ϕ ◦ log (4.13)on a suitable neighborhood of ǫ in exp( U ) .Thus if we identify two local isometries at ǫ that agree on a neighborhood of ǫ , it comesthat a local isometry at ǫ is uniquely determine by its differential at ǫ . A local isometry at ǫ can be uniquely extended to an isometry on G . Indeed, it is well known that if M and N aretwo connected simply connected and geodesically complete pseudo-Riemannian manifolds,then every isometry between connected open subsets of M and N can be uniquely extendedto an isometry between M and N ([45], [67], [71]).Now we have the following theorem which establishes a link between local isometriesat ǫ and the so-called preautomorphisms of the Lie algebra G of G . Theorem 4.2.1. ([64]) Let ( G, µ ) be a connected orthogonal Lie group with Lie algebra G .Let h , i := µ | ǫ stands for the resulting orthogonal structure on G and let P be an endomor-phism of G . Then there exists a local isometry ϕ of G at ǫ with T ǫ ϕ = P if and only if P satisfies1. h P ( x ) , P ( y ) i = h x, y i , for any x, y in G ;2. P (cid:0)(cid:2) x, [ y, z ] (cid:3)(cid:1) = (cid:2) P ( x ) , [ P ( y ) , P ( z )] (cid:3) , for all x, y, z in G . Now, the author quoted above proves that once the Lie algebra of F ( G ) is known onecan readily construct the Lie algebra of I ( G ) . Here is his construction.For g in G we define the following maps : L g : G → G ; R g : G → G ; I g : G → Gh gh h hg h ghg − (4.14)This maps are respectively the left transformation, the right translation and the innerautomorphism by the element g of G . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 i-invariant Metrics on T ∗ G Let ξ be in G and exp : G → G be the exponential map of G . Then, L exp( tξ ) , R exp( tξ ) , I exp( tξ ) are one-parameter subgroups of I ( G ) . Let us note by X ξ,L , X ξ,R and X ξ,I theirrespective infinitesimal generators. Now, we set X ξ,s = X ξ,R + X ξ,L and X ξ,a = X ξ,R − X ξ,L . (4.15)If { α , α , . . . , α m } is a basis of the Lie algebra F ( G ) of F ( G ) and { ξ , ξ , . . . , ξ n } a basisof G then { α , α , . . . , α m , X ξ ,s , X ξ ,s , . . . , X ξ n ,s } is a basis of the Lie algebra I ( G ) .It remains to compute the brackets on I ( G ) . Let us first define some useful objects.Let Paut ( G ) be the group of preautomorphisms of G and denote by F ( G ) the subgroup ofPaut ( G ) consisting of those preautomorphisms which preserve the orthogonal structure on G . Now consider the map T ǫ : F ( G ) → F ( G ) which associates to any element φ of F ( G ) itsdifferential T ǫ φ at the neutral element ǫ of G . The map T ǫ induces a map ∂ ǫ : F ( G ) → F ( G ) between the Lie algebras F ( G ) and F ( G ) of F ( G ) and F ( G ) respectively : ∂ ǫ D = (cid:18) ddt T ǫ [exp I ( G ) ( tD )] (cid:19) | t =0 (4.16)Note that F ( G ) consists of prederivations of G which are skew-symmetric with respect tothe orthogonal structure on G .The following theorem gives the brackets on I ( G ) . Theorem 4.2.2. ([64]) Let ξ , η be in G and D be an element of F ( G ) . Then,1. [ X ξ,s , X η,s ] = [ X ξ,a , X η,a ] = − X [ ξ,η ] ,a ;2. [ D, X ξ,s ] = X ∂ ǫ D ( ξ ) ,s ; Now we conclude that if we know the prederivations of G which are skew-symmetricwith respect to the orthogonal structure on G we can calculate the Lie algebra I ( G ) of thegroup I ( G ) of isometries of the orthogonal Lie group ( G, µ ) . T ∗ G Let G be a Lie group with Lie algebra G . We have already seen (Section 2.2.1) that theduality pairing h , i given by h ( x, f ) , ( y, g ) i = f ( y ) + g ( x ) , (4.17)for all x, y in G , defines an orthogonal structure on the Lie algebra T ∗ G = G ⋉ G ∗ of the Liegroup T ∗ G . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 i-invariant Metrics on T ∗ G T ∗ G Let h , i stand for the duality pairing on T ∗ G , a bi-invariant metric on T ∗ G is given by aninvertible linear map j : T ∗ G → T ∗ G satisfying j ([ u, v ]) = [ u, j ( v )]; (4.18) h j ( u ) , v i = h u, j ( v ) i , (4.19)for all u, v ∈ T ∗ G . Lemma 4.3.1. Any linear map j : T ∗ G → T ∗ G satisfying (4.18) can be written as j ( x, f ) = (cid:16) j ( x ) + j ( f ) , j ( x ) + j ( f ) (cid:17) , (4.20) for any ( x, f ) ∈ T ∗ G ; where j : G → G , j : G → G ∗ , j : G ∗ → G and j : G ∗ → G ∗ arelinear maps such that for all x in G , the following relations hold : j ◦ ad x = ad x ◦ j (4.21) j ◦ ad x = ad ∗ x ◦ j (4.22) j ◦ ad ∗ x = ad x ◦ j = 0 (4.23) (cid:2) j , ad ∗ x (cid:3) = 0 (4.24) ad ∗ x ◦ ( j − j t ) = 0 , (4.25) where j t stands for the transpose of the map j .Proof. If we write u = ( x, f ) and v = ( y, g ) , then j ([ u, v ]) = j (cid:0) [ x, y ] , ad ∗ x g − ad ∗ y f (cid:1) = (cid:16) j ([ x, y ]) + j ( ad ∗ x g − ad ∗ y f ) , j ([ x, y ]) + j ( ad ∗ x g − ad ∗ y f ) (cid:17) (4.26)and [ u, jv ] = [( x, f ) , (cid:0) j ( y ) + j ( g ) , j ( y ) + j ( g ) (cid:1) ]= (cid:16) [ x, j ( y )] + [ x, j ( g )] , ad ∗ x (cid:0) j ( y ) + j ( g ) (cid:1) − ad ∗ j ( y )+ j ( g ) f (cid:17) (4.27)Considering the case where f = g = 0 the equality between (4.27) and (4.28) gives j ([ x, y ]) = [ x, j ( y )] and j ([ x, y ]) = ad ∗ x ( j ( y )) , for all x, y ∈ G . The two relations above are equivalent to (4.21) and (4.22) respectively.The equality between (4.26) and (4.27) now gives (cid:16) j ( ad ∗ x g − ad ∗ y f ) , j ( ad ∗ x g − ad ∗ y f ) (cid:17) = (cid:16) [ x, j ( g )] , ad ∗ x (cid:0) j ( g ) (cid:1) − ad ∗ j ( y )+ j ( g ) f (cid:17) . (4.28)Taking f = 0 , we get on one hand j ( ad ∗ x g ) = [ x, j ( g )] , On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 i-invariant Metrics on T ∗ G for all x ∈ G and all g ∈ G ∗ . Then Relation (4.23) is proved. On the second hand we have j ( ad ∗ x g ) = ad ∗ x ( j ( g )) , for every g in G ∗ ; that is j ◦ ad ∗ x = ad ∗ x ◦ j , (4.29)which is nothing but Relation (4.24). Now if we take x = 0 , we get from Eq. (4.28) (cid:16) j ( − ad ∗ y f ) , j ( − ad ∗ y f ) (cid:17) = (cid:0) , − ad ∗ j ( y )+ j ( g ) f (cid:1) . (4.30)The equality above implies the following two relations j ( − ad ∗ y f ) = 0 ,j ( − ad ∗ y f ) = − ad ∗ j ( y )+ j ( g ) f, for all y ∈ G , f ∈ G ∗ . These equations are equivalent to j ◦ ad ∗ y = 0 , (4.31) j ◦ ad ∗ y = ad ∗ j ( y ) , (4.32) ad ∗ j ( g ) = 0 , (4.33)for all y ∈ G and g ∈ G ∗ . The relations ad ∗ y ◦ j = j ◦ ad ∗ y = ad ∗ j ( y ) = ad ∗ y ◦ j t , (4.34)for all y ∈ G , coming from (4.29) and (4.32), give ad ∗ y ◦ ( j − j t ) = 0 , (4.35)for all y ∈ G ; which means that ( j − j t ) f is a closed -form, for every f ∈ G ∗ . Remark 4.3.1. 1. The equality ad ∗ j ( g ) = 0 , for all g is equivalent to Im ( j ) ⊂ Z ( G ) , (4.36) where Z ( G ) is the centre of G . So if Z ( G ) = { } then j = 0 . 2. Relation (4.34) also gives − h j ( f ) , [ y, x ] i = h j ( ad ∗ y ( f )) , x i = −h f, [ j ( y ) , x ] i = −h f ◦ j , [ y, x ] i , (4.37) for all f in G ∗ and any x, y in G . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 i-invariant Metrics on T ∗ G Any linear map j : T ∗ G → T ∗ G which is h , i -symmetric can be written as j ( x, f ) = (cid:16) j ( x ) + j ( f ) , j ( x ) + j t ( f ) (cid:17) , (4.38) for any ( x, f ) ∈ T ∗ G ; where j : G → G , j : G → G ∗ , j : G ∗ → G are linear maps withthe following relations valid for all x, y in G and all f, g in G ∗ . h j ( x ) , y i = h x, j ( y ) i (4.39) h j ( f ) , g i = h f, j ( g ) i . (4.40) Proof. Let u = ( x, f ) and v = ( y, g ) be two elements of T ∗ G . h ju, v i = D(cid:16) j ( x ) + j ( f ) , j ( x ) + j ( f ) (cid:17) , ( y, g ) E = h j ( x ) , g i + h j ( f ) , g i + h j ( x ) , y i + h j ( f ) , y i (4.41) h u, jv i = D ( x, f ) , (cid:16) j ( y ) + j ( g ) , j ( y ) + j ( g ) (cid:17)E = h f, j ( y ) i + h f, j ( g ) i + h x, j ( y ) i + h x, j ( g ) i (4.42)For f = g = 0 , the equality between (4.41) and (4.42) implies that for all x, y in G , h j ( x ) , y i = h x, j ( y ) i . So (4.39) is established. By the same way, for x = y = 0 , Relation (4.40) is obtained, as h j ( f ) , g i = h f, j ( g ) i . Now the equality between (4.41) and (4.42) can be written h j ( x ) , g i + h j ( f ) , y i = h f, j ( y ) i + h x, j ( g ) i . We take y = 0 to obtain h x, j t ( g ) i = h j ( x ) , g i = h x, j ( g ) i for all x in G and g in G ∗ ; or equivalently j t = j Remark 4.3.2. Relations (4.39) and (4.40) mean that j and j are symmetric withrespect to the duality. Lemma 4.3.3. Let G be a Lie algebra without centre, that is Z ( G ) = { } . A h , i -symmetricbi-invariant endomorphism j : T ∗ G → T ∗ G defined by (4.38) is invertible if and only if j is invertible. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 i-invariant Metrics on T ∗ G Proof. Let j : T ∗ G → T ∗ G be defined by (4.38) with conditions listed in Lemma 4.3.1. Wehave seen in Remark 4.3.1 that if Z ( G ) = { } then j = 0 . The determinant of j is det( j ) = (cid:12)(cid:12)(cid:12)(cid:12) j j j j t (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) j j j t (cid:12)(cid:12)(cid:12)(cid:12) = (cid:0) det( j ) (cid:1) . (4.43)Let us summarize the above three Lemmas in the Proposition 4.3.1. Let G be a Lie algebra. Any h , i -symmetric and adjoint-invariant tensor j : T ∗ G → T ∗ G has the form j ( x, f ) = (cid:16) j ( x ) + j ( f ) , j ( x ) + j t ( f ) (cid:17) , (4.44) for any ( x, f ) ∈ T ∗ G ; where j : G → G , j : G → G ∗ , j : G ∗ → G are linear maps withthe following relations valid for all x, y in G and all f, g in G ∗ . j ◦ ad x = ad x ◦ j (4.45) j ◦ ad x = ad ∗ x ◦ j (4.46) j ◦ ad ∗ x = ad x ◦ j = 0 (4.47) h j ( x ) , y i = h x, j ( y ) i (4.48) h j ( f ) , g i = h f, j ( g ) i . (4.49) If, in addition, the centre Z ( G ) of the Lie algebra is { } , then the endomorphism j isinvertible if and only if j is invertible. T ∗ G In the sequel we will use the duality pairing on T ∗ G and Relation (4.7) in order to determineall non-degenerate symmetric and adjoint-invariant form on T ∗ G , hence all bi-invariantmetrics on T ∗ G . Precisely, any other orthogonal structure µ on T ∗ G is linked to the dualitypairing h , i by µ (cid:16) ( x, f ) , ( y, g ) (cid:17) = D j ( x, f ) , ( y, g ) E , (4.50)for every ( x, f ) , ( y, g ) in T ∗ G ; where j is an endomorphism of T ∗ G such that : • j is invertible; • j is symmetric with respect to the duality, i.e. h j ( u ) , v i = h u, j ( v ) i , for all u, v ∈ T ∗ G ; • j ([ u, v ]) = [ j ( u ) , v ] = [ u, j ( v )] , for all u, v ∈ T ∗ G .We have the following characterization of all orthogonal structures on T ∗ G . Theorem 4.3.1. Let G be a Lie algebra. Any orthogonal structure µ on T ∗ G is given by µ (cid:16) ( x, f ) , ( y, g ) (cid:17) = (cid:10) g, j ( x ) (cid:11) + (cid:10) f, j ( y ) (cid:11) + (cid:10) g, j ( f ) (cid:11) + (cid:10) j ( x ) , y (cid:11) , (4.51) where j : G → G , j : G → G ∗ , j : G ∗ → G are as in Proposition 4.3.1. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 i-invariant Metrics on T ∗ G Proof. Indeed, an orthogonal structure µ on T ∗ G is defined by µ (cid:16) ( x, f ) , ( y, g ) (cid:17) = D j ( x, f ) , ( y, g ) E , where j is given by (4.44) and condition listed in Proposition 4.3.1. We then have µ (cid:16) ( x, f ) , ( y, g ) (cid:17) = D(cid:16) j ( x ) + j ( f ) , j ( x ) + j t ( f ) (cid:17) , ( y, g ) E = D g, j ( x ) + j ( f ) E + D j ( x ) + j t ( f ) , y E = D g, j ( x ) E + D g, j ( f ) E + D j ( x ) , y E + D j t ( f ) , y E = D g, j ( x ) E + D g, j ( f ) E + D j ( x ) , y E + D f, j ( y ) E T ∗ G Let G be a Lie algebra. On T ∗ G we consider the orthogonal structure µ defined by (4.51).We seek to characterise all prederivations of T ∗ G which are skew-symmetric with respectto t he orthogonal structure µ .Let us first recall the space J ′ = { j ′ : G → G linear : [ j ′ , ad x ◦ ad y ] = 0 , ∀ x, y ∈ G } . Proposition 4.3.2. A µ -skew-symmetric prederivation φ of T ∗ G has the form φ ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ( j ′ t − α t )( f ) (cid:17) , (4.52) where j ′ : G → G is in J ′ , α : G → G is a prederivation of G , β : G → G ∗ and ψ : G ∗ → G are as in Theorem 3.3.1, with the additional conditions h β ( x ) , j ( y ) i + h β ( y ) , j ( x ) i + h j ◦ α ( x ) , y i + h j ◦ α ( y ) , x i = 0 (4.53) h f, j ◦ ψ ( g ) i + h g, j ◦ ψ ( f ) i + h f, j ′ ◦ j ( g ) i + h g, j ′ ◦ j ( f ) i − h f, α ◦ j ( g ) i − h g, α ◦ j ( f ) i = 0 (4.54) (cid:2) j , α (cid:3) + j ◦ β + j ′ ◦ j + ψ t ◦ j = 0 (4.55) for every elements x and y of G and any elements f and g in G ∗ .Proof. Recall that according to Corollary 3.3.1 a prederivation φ of T ∗ G can be written as φ ( x, f ) = (cid:16) α ( x ) + ψ ( f ) , β ( x ) + ( j ′ t − α t )( f ) (cid:17) , where j ′ belongs to J ′ and α, β, ψ are as in Theorem 3.3.1. The prederivation φ is µ -skew-symmetric if and only if µ (cid:0) φ ( x, f ) , ( y, g ) (cid:1) + µ (cid:0) ( x, f ) , φ ( y, g ) (cid:1) = 0 , (4.56)for all ( x, f ) and ( y, g ) in T ∗ G . On one hand, we have µ (cid:0) φ ( x, f ) , ( y, g ) (cid:1) = µ (cid:16)(cid:0) α ( x ) + ψ ( f ) , β ( x ) + ( j ′ t − α t )( f ) (cid:1) , ( y, g ) (cid:17) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras = D g , j (cid:0) α ( x ) + ψ ( f ) (cid:1)E + D β ( x ) + ( j ′ t − α t )( f ) , j ( y ) ED g , j (cid:0) β ( x ) + ( j ′ t − α t )( f ) (cid:1)E + D j (cid:0) α ( x ) + ψ ( f ) (cid:1) , y E = (cid:10) g , j ◦ α ( x ) (cid:11) + (cid:10) g , j ◦ ψ ( f ) (cid:11) + (cid:10) β ( x ) , j ( y ) (cid:11) + (cid:10) f , j ′ ◦ j ( y ) (cid:11) − (cid:10) f , α ◦ j ( y ) (cid:11) + (cid:10) g , j ◦ β ( x ) (cid:11) + (cid:10) g , j ◦ j ′ t ( f ) (cid:11) − (cid:10) g , j ◦ α t ( f ) (cid:11) + (cid:10) j ◦ α ( x ) , y (cid:11) + (cid:10) j ◦ ψ ( f ) , y (cid:11) (4.57)On the other hand, µ (cid:0) ( x, f ) , φ ( y, g ) (cid:1) = µ (cid:16) ( x, f ) , (cid:0) α ( y ) + ψ ( g ) , β ( y ) + ( j ′ t − α t )( g ) (cid:1)(cid:17) = D β ( y ) + ( j ′ t − α t )( g ) , j ( x ) E + D f , j (cid:0) α ( y ) + ψ ( g ) (cid:1)E + D β ( y ) + ( j ′ t − α t )( g ) , j ( f ) E + D j ( x ) , α ( y ) + ψ ( g ) E = (cid:10) f , j ◦ α ( y ) (cid:11) + (cid:10) f , j ◦ ψ ( g ) (cid:11) + (cid:10) β ( y ) , j ( x ) (cid:11) + (cid:10) g , j ′ ◦ j ( x ) (cid:11) − (cid:10) g , α ◦ j ( x ) (cid:11) + (cid:10) β ( y ) , j ( f ) (cid:11) + (cid:10) g , j ′ ◦ j ( f ) (cid:11) − (cid:10) g, α ◦ j ( f ) (cid:11) + (cid:10) j ◦ α ( y ) , x (cid:11) + (cid:10) j ◦ ψ ( g ) , x (cid:11) , since j is h , i -symmetric (4.58)Summing (4.57) and (4.58) and taking f = g = 0 , Relation (4.56) becomes (cid:10) β ( x ) , j ( y ) (cid:11) + (cid:10) j ◦ α ( x ) , y (cid:11) + (cid:10) β ( y ) , j ( x ) (cid:11) + (cid:10) j ◦ α ( y ) , x (cid:11) = 0 So (4.53) is proved.Now we rewrite (4.56) and take x = y = 0 to obtain (4.54).Last, (4.55) is obtained by summing (4.57) and (4.58), simplifying and taking y = 0 . Let G be a Lie algebra. We recall that an orthogonal structure µ on G induces an isomor-phism θ : G → G ∗ by the formula h θ ( x ) , y i = µ ( x, y ) , for all x, y in G . We also recall the set J = { j : G → G : j ([ x, y ]) = [ j ( x ) , y ] , ∀ x, y ∈ G } . Lemma 4.4.1. Let ( G , µ ) be an orthogonal Lie algebra. Any bi-invariant h , i -symmetricinvertible endomorphism j : T ∗ G → T ∗ G is given by j ( x, f ) = (cid:16) j ( x ) + j ◦ θ − ( f ) , θ ◦ j ( x ) + j t ( f ) (cid:17) , (4.59) for any ( x, f ) in T ∗ G ; where j , j , j are elements of J such that • j is invertible, On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras • j is µ -symmetric • and j satisfies the following two relations j ◦ ad x = 0 , (4.60) h j ◦ θ − ( f ) , g i = h f , j ◦ θ − ( g ) i (4.61) for all x in G and all f, g in G ∗ .Proof. We have already seen that any bi-invariant h , i -symmetric invertible endomorphismof T ∗ G is given by j ( x, f ) = (cid:16) j ( x ) + j ( f ) , j ( x ) + j t ( f ) (cid:17) , with conditions listed in Proposition 4.3.1.The linear maps θ − ◦ j : G → G and j ◦ θ : G → G commutes with all adjointoperators of G ; that is there exists j , j in J such that θ − ◦ j = j and j ◦ θ = j . Itcomes that j = θ ◦ j and j = j ◦ θ − .Now, (4.47) is equivalent to j ◦ θ − ◦ ad ∗ x = 0 , for all x in G . Since θ − ◦ ad ∗ x = ad x ◦ θ − ,for all x in G , we have j ◦ ad x ◦ θ − = 0 , for any x in G . It comes that j ◦ ad x = 0 , for all x in G , since θ − is an isomorphism.Relation (4.48) becomes h θ ◦ j ( x ) , y i = h x, θ ◦ j ( y ) i , for any x, y in G . The latter can bewritten µ (cid:0) j ( x ) , y (cid:1) = µ (cid:0) x, j ( y ) (cid:1) , for any x, y in G .Last, Relation (4.49) reads h j ◦ θ − ( f ) , g i = h f, j ◦ θ − ( g ) i , for all f, g in G ∗ . Remark 4.4.1. The relation j ◦ ad x = 0 , for all x in G means that j vanishes identicallyon the derived ideal [ G , G ] of G . So, if G is perfect, then j vanishes identically on all G . The following comes immediately from Lemma 4.4.1 and Remark 4.4.1 Corollary 4.4.1. Let ( G , µ ) be an orthogonal and perfect Lie algebra. Then, any bi-invariant h , i -symmetric invertible endomorphism j : T ∗ G → T ∗ G is given by j ( x, f ) = (cid:16) j ( x ) , θ ◦ j ( x ) + j t ( f ) (cid:17) , (4.62) for any ( x, f ) in T ∗ G ; where j , j are bi-invariant tensors G such that j is µ -symmetricand j is invertible. Theorem 4.4.1. Let ( G , µ ) be an orthogonal Lie algebra. Any orthogonal structure µ D on T ∗ G is given by µ D (cid:16) ( x, f ) , ( y, g ) (cid:17) = (cid:10) g , j ( x ) (cid:11) + (cid:10) f , j ( y ) (cid:11) + (cid:10) g, j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) , y (cid:11) , (4.63) for all ( x, f ) , ( y, g ) in T ∗ G , where j , j and j satisfy conditions listed in Lemma 4.4.1. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras Proof. Any orthogonal structure µ D on T ∗ G is linked to the duality pairing by means ofa map j : T ∗ G → T ∗ G defined by (4.59) through the following formula valid for all ( x, f ) , ( y, g ) in T ∗ G : µ D (cid:16) ( x, f ) , ( y, g ) (cid:17) = h j ( x, f ) , ( y, g ) i . = D(cid:16) j ( x ) + j ◦ θ − ( f ) , θ ◦ j ( x ) + j t ( f ) (cid:17) , ( y, g ) E = (cid:10) g , j ( x ) + j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) + j t ( f ) , y (cid:11) = (cid:10) g , j ( x ) (cid:11) + (cid:10) g , j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) , y (cid:11) + (cid:10) j t ( f ) , y (cid:11) = (cid:10) g , j ( x ) (cid:11) + (cid:10) g , j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) , y (cid:11) + (cid:10) f , j ( y ) (cid:11) We have the following consequence coming from Theorem 4.4.1 and Remark 4.4.1. Corollary 4.4.2. Let ( G , µ ) be an orthogonal and perfect Lie algebra. Any orthogonalstructure µ D on T ∗ G is given by µ D (cid:16) ( x, f ) , ( y, g ) (cid:17) = (cid:10) g , j ( x ) (cid:11) + (cid:10) f , j ( y ) (cid:11) + (cid:10) θ ◦ j ( x ) , y (cid:11) , (4.64) for all ( x, f ) , ( y, g ) in T ∗ G , where j and j are are bi-invariant tensors of G such that j is invertible, j is µ -symmetric. Let us now characterise skew-symmetric prederivations of T ∗ G in the case where G isan orthogonal Lie algebra. Proposition 4.4.1. Let ( G , µ ) be an orthogonal Lie algebra. Then any prederivation φ of T ∗ G which is skew-symmetric with respect to the orthogonal structure µ D defined by (4.63)can be written as φ ( x, f ) = (cid:16) α ( x ) + j ′ ◦ θ − ( f ) , θ ◦ α ( x ) + ( j ′ t − α t )( f ) (cid:17) , (4.65) where α , α are in Pder ( G ) ; j ′ , j ′ are in J ′ with the following additional conditions: θ ◦ j ◦ α + α t ◦ θ ◦ j + α t ◦ θ ◦ j = 0 (4.66) ( j ◦ j ′ + j ′ ◦ j − α ◦ j ) ◦ θ − + j ◦ θ − ◦ ( j ′ t − α t ) + θ − t ◦ j ′ t ◦ j t = 0 (4.67) (cid:2) j , α (cid:3) + j ◦ α + j ′ ◦ j + θ − t ◦ j ′ t ◦ θ ◦ j = 0 , (4.68) where θ − t is the transpose of θ − .Proof. From Proposition 3.4.1, if G is an orthogonal Lie algebra, then any prederivation of T ∗ G is given by φ ( x, f ) = (cid:16) α ( x ) + j ′ ◦ θ − ( f ) , θ ◦ α ( x ) + ( j ′ t − α t )( f ) (cid:17) , On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Orthogonal Lie Algebras for every ( x, f ) in T ∗ G , where α , α are prederivations of G , j ′ , j ′ are in J ′ . The prederiva-tion φ is µ D -skew-symmetric means that µ D (cid:16) φ ( x, f ) , ( y, g ) (cid:17) + µ D (cid:16) ( x, f ) , φ ( y, g ) (cid:17) = 0 , (4.69)for all ( x, f ) , ( y, g ) in T ∗ G . Firstly, we compute µ D (cid:0) φ ( x, f ) , ( y, g ) (cid:1) . µ D (cid:16) φ ( x, f ) , ( y, g ) (cid:17) = µ D (cid:16)(cid:0) α ( x ) + j ′ ◦ θ − ( f ) , θ ◦ α ( x ) + ( j ′ t − α t )( f ) (cid:1) , ( y, g ) (cid:17) = D g , j (cid:16) α ( x ) + j ′ ◦ θ − ( f ) (cid:17)E + h f, j ( y ) i + D g , j ◦ θ − (cid:16) θ ◦ α ( x ) + ( j ′ t − α t )( f ) (cid:17)E + D θ ◦ j (cid:16) α ( x ) + j ′ ◦ θ − ( f ) (cid:17) , y E = D g , j ◦ α ( x ) E + D g , j ◦ j ′ ◦ θ − ( f ) E + h f, j ( y ) i + D g , j ◦ α ( x ) E + D g , j ◦ θ − ◦ j ′ t ( f ) E − D g , j ◦ θ − ◦ α t ( f ) E + D θ ◦ j ◦ α ( x ) , y E + D θ ◦ j ◦ j ′ ◦ θ − ( f ) , y E . (4.70)Secondly, µ D (cid:16) ( x, f ) , φ ( y, g ) (cid:17) = µ D (cid:16) ( x, f ) , (cid:0) α ( y ) + j ′ ◦ θ − ( g ) , θ ◦ α ( y ) + ( j ′ t − α t )( g ) (cid:1)(cid:17) = D θ ◦ α ( y )+( j ′ t − α t )( g ) , j ( x ) E + D f , j (cid:0) α ( y )+ j ′ ◦ θ − ( g ) (cid:1)E + D θ ◦ α ( y ) + ( j ′ t − α t )( g ) , j ◦ θ − ( f ) E + D θ ◦ j ( x ) , α ( y ) + j ′ ◦ θ − ( g ) E = D θ ◦ α ( y ) , j ( x ) E + D j ′ t ( g ) , j ( x ) E − D α t ( g ) , j ( x ) E + D f , j ◦ α ( y ) E + D f , j ◦ j ′ ◦ θ − ( g ) E + D θ ◦ α ( y ) , j ◦ θ − ( f ) E + D j ′ t ( g ) , j ◦ θ − ( f ) E − D α t ( g ) , j ◦ θ − ( f ) E + D θ ◦ j ( x ) , α ( y ) E + D θ ◦ j ( x ) , j ′ ◦ θ − ( g ) E . (4.71)Now, if we take f = g = 0 , (4.69) gives D θ ◦ j ◦ α ( x ) , y E + D θ ◦ α ( y ) , j ( x ) E + D θ ◦ j ( x ) , α ( y ) E = 0 which can be written D θ ◦ j ◦ α ( x ) , y E + D y , α t ◦ θ ◦ j ( x ) E + D α t ◦ θ ◦ j ( x ) , y E = 0 for all x, y in G . Then, θ ◦ j ◦ α + α t ◦ θ ◦ j + α t ◦ θ ◦ j = 0 . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Semi-simple Lie Algebras We can rewrite (4.69) as follows (cid:10) g , j ◦ α ( x ) (cid:11) + (cid:10) g , j ◦ j ′ ◦ θ − ( f ) (cid:11) + h f, j ( y ) i + (cid:10) g , j ◦ α ( x ) (cid:11) + (cid:10) g , j ◦ θ − ◦ j ′ t ( f ) (cid:11) − (cid:10) g , j ◦ θ − ◦ α t ( f ) (cid:11) + (cid:10) θ ◦ j ◦ j ′ ◦ θ − ( f ) , y (cid:11) + (cid:10) j ′ t ( g ) , j ( x ) (cid:11) − (cid:10) α t ( g ) , j ( x ) (cid:11) + (cid:10) f , j ◦ α ( y ) (cid:11) + (cid:10) f , j ◦ j ′ ◦ θ − ( g ) (cid:11) + (cid:10) θ ◦ α ( y ) , j ◦ θ − ( f ) (cid:11) + (cid:10) j ′ t ( g ) , j ◦ θ − ( f ) (cid:11) − (cid:10) α t ( g ) , j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) , j ′ ◦ θ − ( g ) (cid:11) = 0 . (4.72)If we take x = y = 0 , (4.72) becomes D g , j ◦ j ′ ◦ θ − ( f ) E + D g , j ◦ θ − ◦ j ′ t ( f ) E − D g , j ◦ θ − ◦ α t ( f ) E + D f , j ◦ j ′ ◦ θ − ( g ) E + D j ′ t ( g ) , j ◦ θ − ( f ) E − D α t ( g ) , j ◦ θ − ( f ) E = 0 (4.73)which is D g , j ◦ j ′ ◦ θ − ( f ) E + D g , j ◦ θ − ◦ j ′ t ( f ) E − D g , j ◦ θ − ◦ α t ( f ) E + D θ − t ◦ j ′ t ◦ j t ( f ) , g E + D g , j ′ ◦ j ◦ θ − ( f ) E − D g , α ◦ j ◦ θ − ( f ) E = 0 , for all f, g in G ∗ , where θ − t is the transpose of θ − . This latter equality is equivalent to ( j ◦ j ′ + j ′ ◦ j − α ◦ j ) ◦ θ − + j ◦ θ − ◦ ( j ′ t − α t ) + θ − t ◦ j ′ t ◦ j t = 0 Last, (4.69) have a more simple expression : (cid:10) g , j ◦ α ( x ) (cid:11) + h f, j ( y ) i + (cid:10) g , j ◦ α ( x ) (cid:11) + (cid:10) θ ◦ j ◦ j ′ ◦ θ − ( f ) , y (cid:11) + (cid:10) j ′ t ( g ) , j ( x ) (cid:11) − (cid:10) α t ( g ) , j ( x ) (cid:11) + (cid:10) f , j ◦ α ( y ) (cid:11) + (cid:10) θ ◦ α ( y ) , j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) , j ′ ◦ θ − ( g ) (cid:11) = 0 . (4.74)We take y = 0 and obtain h g , j ◦ α ( x ) i + h g , j ◦ α ( x ) i + h g , j ′ ◦ j ( x ) i−h g , α ◦ j ( x ) i + h θ ◦ j ( x ) , j ′ ◦ θ − ( g ) i = 0 (4.75)The latter equality is equivalent to [ j , α ] + j ◦ α + j ′ ◦ j + θ − t ◦ j ′ t ◦ θ ◦ j = 0 . Let G = s ⊕ s ⊕ · · · ⊕ s p be a semi-simple Lie algebra, where s i , i = 1 , , . . . , p ( p ∈ N ∗ )are simple ideals. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Semi-simple Lie Algebras Lemma 4.5.1. Any h , i -symmetric and invertible bi-invariant tensor j : T ∗ G → T ∗ G isdefined by j ( x, f ) = (cid:16) p X i =1 λ i x i , p X k =1 ν k θ ( x k ) + p X i =1 λ i f i (cid:17) , (4.76) where x = x + x + · · · + x p ∈ s ⊕ s ⊕ · · · ⊕ s p , f = f + f + · · · + f p ∈ s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p , λ i , ( i = 1 , , . . . , p ) are non-zero real numbers, ν i ( i = 1 , , . . . , p ) are any real numbersand θ : G → G ∗ is the isomorphism induced by any orthogonal structure on G through theformula (1.2).Proof. According to Corollary 4.4.1, any bi-invariant, h , i -symmetric and invertible endo-morphism j : T ∗ G → T ∗ G has the form j ( x, f ) = (cid:16) j ( x ) , θ ◦ j ( x ) + j t ( f ) (cid:17) , (4.77)for all ( x, f ) in T ∗ G , where j and j are bi-invariant endomorphisms of G such that • j is invertible, • j is symmetric with respect to any orthogonal structure on G .Now we have seen in Section 2.4.1 that the map j must have the form j ( x ) = p X i =1 λ i x i , (4.78)if x = x + x + · · · + x p ∈ s ⊕ s ⊕ · · · ⊕ s p , where λ i , i = 1 , , . . . , p are real numbers. Itcomes that j t ( f ) = p X i =1 λ i f i , (4.79)if f = f + f + · · · + f p ∈ s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p .We also have j ( x ) = p X k =1 ν k x k , (4.80)where ν k , k = 1 , , . . . , p , are real numbers. So we have θ ◦ j ( x ) = θ (cid:16) p X k =1 ν k x k (cid:17) = p X k =1 ν k θ ( x k ) (4.81)We then can write j as follows j ( x, f ) = (cid:16) p X i =1 λ i x i , p X k =1 ν k θ ( x k ) + p X i =1 λ i f i (cid:17) (4.82) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Semi-simple Lie Algebras Now j is invertible if and only if j = P pi =1 λ i Id s i is invertible.For all i = 1 , , . . . , p , note by n i the dimension of the simple ideal s i ( n i := dim s i ).The matrix of j , in some basis { e , e , . . . , e n ; e , e , . . . , e n ; . . . ; e p , e p , . . . , e pn p } ,is the following A O . . . OO A . . . O ... ... . . . OO O . . . A p (4.83)where A i is the n i × n i -matrix defined as follows A i = diag ( λ i , λ i , . . . , λ i | {z } n i times ) = λ i O . . . OO λ i . . . O ... ... . . . OO O . . . λ i (4.84)The determinant of j is then det( j ) = p Y k =1 λ n k k . (4.85)Then, j is invertible if and only if each of λ k is non-zero. Remark 4.5.1. The matrix of j in the basis { e , e , . . . , e n ; e , e , . . . , e n ; . . . ; e p , e p , . . . , e pn p ; e ∗ , e ∗ , . . . , e ∗ n ; e ∗ , e ∗ , . . . , e ∗ n ; . . . ; e ∗ p , e ∗ p , . . . , e ∗ pn p } is given by A O . . . O O O . . . OO A . . . O O O . . . O ... ... . . . O ... ... ... ... O O . . . A p O O . . . OA O . . . OO A . . . O ... ... . . . ... O O . . . A p (4.86) and the determinant of j is det( j ) = p Y k =1 λ n k k . (4.87)Note by h , i s i the duality pairing between s i and its dual space s ∗ i . The following theoremcharacterizes all orthogonal structures on T ∗ G . Theorem 4.5.1. Let G be a semi-simple Lie algebra. Any orthogonal structure µ on T ∗ G is given by µ (cid:0) ( x, f ) , ( y, g ) (cid:1) = p X i =1 λ i (cid:10) ( x i , f i ) , ( y i , g i ) (cid:11) s i + p X k =1 ν k h θ ( x k ) , y k i s k , (4.88) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Semi-simple Lie Algebras for all x, y in G = s ⊕ s ⊕ · · · ⊕ s p and f, g in G ∗ = s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p ; where for all i = 1 , , . . . , p , ( λ i , ν i ) belongs to R ∗ × R .In particular, if G is simple, then any orthogonal structure can be written µ (cid:0) ( x, f ) , ( y, g ) (cid:1) = λ (cid:10) ( x, f ) , ( y, g ) (cid:11) + ν h θ ( x ) , y i , (4.89) for all ( x, f ) , ( y, g ) in T ∗ G , where ( λ, ν ) is in R ∗ × R .Proof. Any orthogonal structure on T ∗ G is given by an adjoint-invariant invertible and h , i -symmetric endomorphism j through the formula µ j (cid:0) ( x, f ) , ( y, g ) (cid:1) = h j ( x, f ) , ( y, g ) i , (4.90)We have seen in Lemma 4.5.1 that j ( x, f ) = (cid:16) p X i =1 λ i x i , p X k =1 ν k θ ( x k ) + p X i =1 λ i f i (cid:17) , where x = x + x + · · · + x p ∈ s ⊕ s ⊕ · · · ⊕ s p , f = f + f + · · · + f p ∈ s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p , λ i , ν i , ( i = 1 , , . . . , p ) are real numbers, λ i = 0 , for all i = 1 , , . . . , p . We can write µ j (cid:0) ( x, f ) , ( y, g ) (cid:1) = p X i =1 λ i g ( x i ) + p X k =1 ν k θ ( x k )( y ) + p X i =1 λ i f i ( y )= p X i =1 λ i (cid:16) g ( x i ) + f i ( y ) (cid:17) + p X k =1 ν k θ ( x k )( y )= p X i =1 λ i (cid:16) g i ( x i ) + f i ( y i ) (cid:17) + p X k =1 ν k θ ( x k )( y k )= p X i =1 λ i (cid:10) ( x i , f i ) , ( y i , g i ) (cid:11) s i + p X k =1 ν k h θ ( x k ) , y k i s k (4.91)So, the first part of the theorem is proved.Now, if G is simple, then G = s , x = x , f = f and Relation (4.91) becomes µ j (cid:0) ( x, f ) , ( y, g ) (cid:1) = λ (cid:10) ( x, f ) , ( y, g ) (cid:11) + ν h θ ( x ) , y i (4.92)where λ ∈ R ∗ and ν ∈ R . Remark 4.5.2. If G = s ⊕ s ⊕· · ·⊕ s p is a semi-simple Lie algebra, where s i , i = 1 , , . . . , p are simple ideals, then on any of the ideals s i the Killing form K s i defines an orthogonalstructure. Thus, Relation (4.88) can be written µ (cid:0) ( x, f ) , ( y, g ) (cid:1) = p X i =1 λ i (cid:10) ( x i , f i ) , ( y i , g i ) (cid:11) s i + p X k =1 ν k K s k ( x k , y k ) (4.93) and Relation (4.89) can be expressed as µ j (cid:0) ( x, f ) , ( y, g ) (cid:1) = λ (cid:10) ( x, f ) , ( y, g ) (cid:11) + νK ( x, y ) , (4.94) where K stands for the Killing form on G . Let us now study the group of isometries of bi-invariant metrics on T ∗ G , when G is asemi-simple Lie group. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ase of Semi-simple Lie Algebras Proposition 4.5.1. Let G be a semi-simple Lie. Then any prederivation of T ∗ G which isskew-symmetric with respect to any orthogonal structure µ on T ∗ G is an inner derivationof T ∗ G ; that is if φ is a µ -skew-symmetric prederivation of T ∗ G , then there exist x in G and f in G ∗ such that φ ( x, f ) = ([ x , x ] , ad ∗ x f − ad ∗ x f ) , (4.95) for every x and f in G and G ∗ respectively.Proof. We have shown in Theorem 3.4.1 that if G is a semi-simple Lie algebra, then everyprederivation of T ∗ G is a derivation. From Relation (2.62) we have that, if G is semi-simple,any derivation of T ∗ G has the form φ ( x, f ) = ([ x , x ] , ad ∗ x f − ad ∗ x f + p X i =1 γ i f i ) (4.96)for every x in G and every f := f + f + · · · + f p in s ∗ ⊕ s ∗ ⊕ · · · ⊕ s ∗ p = G ∗ , where x is in G , f is in G ∗ and γ i , i = 1 , .., p , are real numbers. To prove the Proposition it suffices toprove that φ is µ -skew-symmetric if and only if γ i = 0 , for all i = 1 , , . . . , p . µ (cid:16) φ ( x, f ) , ( y, g ) (cid:17) = µ (cid:16)(cid:0) [ x , x ] , ad ∗ x f − ad ∗ x f + p X i =1 γ i f i (cid:1) , ( y, g ) (cid:17) = p X i =1 λ i D(cid:16) [ x , x ] i , (cid:0) ad ∗ x f − ad ∗ x f + p X k =1 γ k f k (cid:1) i (cid:17) , ( y i , g i ) E s i + p X k =1 ν k D θ ( y k ) , [ x , x ] k E s k That is µ (cid:16) φ ( x, f ) , ( y, g ) (cid:17) = p X i =1 λ i " g i ([ x , x ] i ) + ( ad ∗ x f ) i ( y i ) − ( ad ∗ x f ) i ( y i ) + p X k =1 γ k f k ! i ( y i ) + p X k =1 ν k h θ ( y k ) , [ x , x ] k i (4.97)By the same way we obtain µ (cid:16) ( x, f ) , φ ( y, g ) (cid:17) = p X i =1 λ i " f i ([ x , y ] i ) + ( ad ∗ x g ) i ( x i ) − ( ad ∗ y f ) i ( x i ) + p X k =1 γ k g k ! i ( x i ) + p X k =1 ν k h θ ([ x , y ] k ) , x k i . (4.98)Now φ is µ -symmetric means that µ (cid:16) φ ( x, f ) , ( y, g ) (cid:17) + µ (cid:16) ( x, f ) , φ ( y, g ) (cid:17) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples = − p X i =1 λ i h ( ad ∗ x f ) i ( y i ) + ( ad ∗ y f )( x i ) i + p X k =1 ν k h h θ ( y k ) , [ x , x ] k i + h θ ([ x , y ] k ) , x k i i + p X i =1 λ i g i ([ x , x ] i ) + p X i =1 λ i ( ad ∗ x f ) i ( y i ) + p X i =1 λ i p X k =1 γ k f k ! i ( y i )+ p X i =1 λ i f i ([ x , y ] i ) + p X i =1 λ i ( ad ∗ x g ) i ( x i ) + p X i =1 λ i p X k =1 γ k g k ! i ( x i )= p X i =1 λ i f (cid:16) [ x i , y i ] + [ y i , x i ] | {z } =0 (cid:17) + p X k =1 ν k h h θ ( y k ) , [ x , x k ] i − h θ ( y k ) , [ x , x k ] | {z } =0 i i + p X i =1 λ i g i ([ x , x ] i ) + ( ad ∗ x g ) i ( x i ) | {z } =0 + p X i =1 f ([ x , y ] i ) + ( ad ∗ x f ) i ( y i ) | {z } =0 + X i =1 λ i " p X k =1 γ k f k ! i ( y i ) + p X k =1 γ k g k ! i ( x i ) . (4.99)We then have p X i =1 λ i " p X k =1 γ k f k ! i ( y i ) + p X k =1 γ k g k ! i ( x i ) = 0 p X i =1 λ i h γ i f i ( y i ) + γ i g i ( x i ) i = 0 (4.100)for all x, y in G and all f, g in G ∗ . Now if we take y = 0 , we obtain p X i =1 λ i p X k =1 γ k g k ! i ( x i ) = 0 , (4.101)for all x, y in G and all f, g in G ∗ . That is p X i =1 λ i γ i g i ( x i ) = 0 , (4.102)for all x = x + x + · · · + x p ∈ G and for all g = g + g + · · · + g p in G ∗ . Since, λ i = 0 , forany i = 1 , , . . . , p , then we have γ i = 0 , for any i = 1 , , . . . , p . R Let G := Aff ( R ) be the Lie group of affine motions of R and let G := aff ( R ) its Lie algebra.We note by T ∗ G := T ∗ aff ( R ) = G ⋉ G ∗ the Lie algebra of the cotangent bundle of the Liegroup G . If G := span { e , e } and { e , e } denotes the dual basis of { e , e } , we have thefollowing brackets: [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e (4.103) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples Let h , i denote the duality pairing between G and G ∗ .1. Any orthogonal structure µ on T ∗ G is given by the following expression: µ (cid:16) ( x, f ) , ( y, g ) (cid:17) = a h ( x, f ) , ( y, g ) i + bx y , (4.104) for all ( x, f ) = x e + x e + f e + f e and ( y, g ) = y e + y e + g e + g e in T ∗ G ,where a ∈ R ∗ and b ∈ R .2. Any orthogonal structure µ on T ∗ G is of signature (2 , .Proof. On the basis ( e , e , e , e ) of T ∗ G an invertible bi-invariant tensor j : T ∗ G → T ∗ G has the following matrix a a b a 00 0 0 a , a ∈ R ∗ , b ∈ R . (4.105)We note j a,b := j if j is defined by (4.105). One can verify that j a,b is h , i -symmetric.Now any adjoint-invariant scalar product on T ∗ G is given by an adjoint-invariant invertibleand h , i -symmetric endomorphism j a,b . We note it by µ j a,b or simply by µ a,b . We have µ a,b (cid:16) ( x, f ) , ( y, g ) (cid:17) = h j a,b ( x, f ) , ( y, g ) i , (4.106)for all ( x, f ) and ( y, g ) in T ∗ G . Now if we set ( x, f ) = x e + x e + f e + f e and ( y, g ) = y e + y e + g e + g e , then we have µ a,b (cid:16) ( x, f ) , ( y, g ) (cid:17) = (cid:10) ax e + ax e + ( bx + af ) e + af , y e + y e + g e + g e (cid:11) = ax g + ax g + ( bx + af ) y + ay f = a ( x g + x g + y f + y f ) + bx y . That is µ a,b (cid:16) ( x, f ) , ( y, g ) (cid:17) = a h ( x, f ) , ( y, g ) i + bx y , for all ( x, f ) and ( y, g ) in T ∗ G , where a ∈ R ∗ and b ∈ R .Let us study the signature of the scalar product µ a,b . The matrix of µ a,b on the basis ( e , e , e , e ) of T ∗ G is given by M a,b = b a 00 0 0 aa a . (4.107)The characteristic polynomial of M a,b is P a,b ( t ) = ( t − a )( t + a )( t − bt − a ) , (4.108) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples for any t ∈ R . It is now a little matter to check that M a,b has four eigenvalues : − a , a , b − √ b + 4 a , b + √ b + 4 a . Two of these eigenvalues are less than zero and two are greater than zero. Hence, µ a,b is ofsignature (2 , for all ( a, b ) ∈ R ∗ × R .Let G stand for the identity connected component of G . We also note by µ a,b thebi-invariant metric induced on the Lie group T ∗ G by the orthogonal structure µ a,b givenby (4.104). Proposition 4.6.2. Any µ a,b -skew-symmetric prederivation φ of T ∗ G is an inner derivationof T ∗ G .Proof. A prederivation α of T ∗ G has the following matrix on the basis ( e , e , e , e ) of T ∗ G (see Examples 2.5.1 and 3.5.1): α α α α α α − α − α α − α (4.109)A µ a,b -skew-symmetric prederivation α of T ∗ G is characterised by µ a,b (cid:16) α ( x, f ) , ( y, g ) (cid:17) = − µ a,b (cid:16) ( x, f ) , α ( y, g ) (cid:17) , (4.110)for all ( x, f ) and ( y, g ) in T ∗ G . The left hand side of the equality above is µ a,b (cid:16) α ( x, f ) , ( y, g ) (cid:17) = µ a,b (cid:16) ( α x + α x + α f ) e +( α x + α x + α f − α f ) e +( − α x +( α − α ) f ) e , y e + y e + g e + g e (cid:17) = a D ( α x + α x + α f ) e +( α x + α x + α f − α f ) e +( − α x +( α − α ) f ) e , y e + y e + g e + g e E = a h ( α x + α x + α f ) g +( α x + α x + α f − α f ) y +( − α x +( α − α ) f ) y i = a h α ( x g − y f ) + α x g + α f g + α x y + α ( x y − x y ) + α y f + α x y + α y f i (4.111)The right hand side gives − µ a,b (cid:16) ( x, f ) , α ( y, g ) (cid:17) = − µ a,b (cid:16) x e + x e + f e + f e , ( α y + α y + α g ) e +( α y + α y + α g − α g ) e +( − α y + α y + α g ) e (cid:17) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples = − a h ( α y + α y + α g ) f + x ( α y + α y + α g − α g )+ x ( − α y + α y + α g ) i = − a h α ( − x g + y f ) + α y f + α f g + α x y + α ( − x y + x y ) + α x g + α x y + α x g i (4.112)The equality (4.110) then implies that α = α = α = 0 . In this case α can bewritten α = α φ + α φ + α φ . One can readily check that φ = − ad e , φ = ad e , φ = − ad e .Let us now focus our attention on isometries of bi-invariant metrics on T ∗ G . Recallfirst the following materials. • The operation law of T ∗ G is given by (see Proposition 5.3.1) : x · y = (cid:16) x + y , x + y e x , x + y + x y e − x , x + y e − x (cid:17) . (4.113)The unit element is ǫ = (0 , , , and the inverse of an element x = ( x , x , x , x ) is the element x − = ( − x , − x e − x , − x + x x , − x e x ) . • The exponential map of T ∗ G is defined as follows (see Chapter 5, Corollary 5.3.1) :let ξ = ξ e + ξ e + ξ e + ξ e be in T ∗ G , exp( ξ )= (cid:18) , ξ , ξ ξ + ξ , ξ (cid:19) , if ξ = 0 (cid:18) ξ , ξ ξ [exp( ξ ) − , ξ + ξ ξ ξ + ξ ξ ξ [exp( − ξ ) − , ξ ξ [1 − exp( − ξ )] (cid:19) if ξ = 0 . For i = 1 , , , , note by X iL , X iR the infinitesimal generators of the one-parameter sub-groups L exp ( te i ) and R exp ( te i ) respectively. It is readily checked that for all g = ( x , x , x , x ) in T ∗ G , we have : X ,L | g = (1 , x , , − x ) , X ,R | g = (1 , , , X ,L | g = (0 , , x , , X ,R | g = (0 , e x , , X ,L | g = (0 , , , , X ,R | g = (0 , , , X ,L | g = (0 , , , , X ,R | g = (0 , , x e − x , . (4.114) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples It is now easy to obtain X ,s | g = (2 , x , , − x ) , X ,a | g = (0 , − x , , x ) X ,s | g = (0 , e x , x , , X ,a | g = (0 , e x − , − x , X ,s | g = (0 , , , , X ,a | g = (0 , , , X ,s | g = (0 , , x e − x , e − x ) , X ,a | g = (0 , , x e − x , e − x − (4.115)Now { φ , φ , φ , X ,s , X ,s , X ,s , X ,s } is a basis of the Lie algebra I ( G ) of the Lie group I ( G ) of isometries of any bi-invariant metric on T ∗ G . Now we just have to compute thebrackets on I ( G ) by Theorem 4.2.2. The non-vanishing brackets are the following : [ φ , φ ] = − φ , [ φ , X ,s ] = X ,s , [ φ , X ,s ] = − X ,s [ φ , X ,s ] = X ,s , [ φ , X ,s ] = − X ,s , [ φ , X ,s ] = X ,s [ φ , X ,s ] = − X ,s , [ X ,s , X ,s ] = − X ,s , [ X ,s , X ,s ] = X ,s [ X ,s , X ,s ] = − X ,s . (4.116) SL (2 , R ) Let G denote the special linear group SL (2 , R ) . Set G := sl (2 , R ) = span { e , e , e } and T ∗ G := G ⋉ G ∗ = span { e , e , e , e , e , e } . We have the following brackets (see Example2.5.3): [ e , e ] = − e [ e , e ] = 2 e [ e , e ] = 2 e [ e , e ] = − e [ e , e ] = − e [ e , e ] = e [ e , e ] = − e [ e , e ] = − e [ e , e ] = 2 e (4.117)An element ( x, f ) of T ∗ G can be written ( x, f ) = x e + x e + x e + f e + f e + f e . Proposition 4.6.3. Let h , i stand for the duality pairing between G and G ∗ and µ be anyorthogonal structure on T ∗ G . Then,1. there exist ( a, b ) in R ∗ × R such that µ (cid:0) ( x, f ) , ( y, g ) (cid:1) = a (cid:10) ( x, f ) , ( y, g ) (cid:11) + 4 b (2 x y + x y + x y ) , (4.118) for all ( x, f ) and ( y, g ) in T ∗ G .2. The orthogonal structure µ on T ∗ G is of signature (3 , .Proof. 1. Since G is a simple Lie algebra, Theorem 4.5.1 asserts that any orthogonalstructure µ a,b on T ∗ G is given by µ a,b (cid:0) ( x, f ) , ( y, g ) (cid:1) = a (cid:10) ( x, f ) , ( y, g ) (cid:11) + b h θ ( x ) , y i , (4.119)for all ( x, f ) , ( y, g ) in T ∗ G , where ( a, b ) ∈ R ∗ × R . Because of the simplicity of G every orthogonal structure on G is a multiple of the Killing form. So, we let θ beinduced by the Killing form K of G , i.e. h θ ( x ) , y i = K ( x, y ) , for all x, y in G . Then, µ (cid:0) ( x, f ) , ( y, g ) (cid:1) = a (cid:10) ( x, f ) , ( y, g ) (cid:11) + bK ( x, y ) , On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples for all ( x, f ) , ( y, g ) in T ∗ G .By definition, the Killing form is given by K ( x, y ) = trace ( ad x ◦ ad y ) , (4.120)for all x, y in G . On the basis of ( e , e , e ) of G , we have the following matrix of K Now it is a little matter to checked that µ a,b (cid:0) ( x, f ) , ( y, g ) (cid:1) = a (cid:10) ( x, f ) , ( y, g ) (cid:11) + b (8 x y + 4 x y + 4 x y ) 2. The matrix of µ a,b on the basis of T ∗ G is M a,b = b a b a 00 4 b aa a a . (4.121)The characteristic polynomial of M a,b is P a,b ( t ) = ( t − bt − a ) (cid:2) t − a + 8 b ) t + a (cid:3) , (4.122)for all t ∈ R . The roots of P a,b are t = 4 b − √ a + 16 b ; t = 4 b + √ a + 16 b t = − q a + 8 b − | b |√ a + 4 b ; t = + q a + 8 b − | b |√ a + 4 b t = − q a + 8 b + 4 | b |√ a + 4 b ; t = + q a + 8 b + 4 | b |√ a + 4 b The roots t , t and t are negative while t , t and t are positive. Then, the signatureof the bilinear form µ a,b is (3 , , for all ( a, b ) ∈ R ∗ × R .From Theorem 3.4.1, any prederivation of T ∗ G is a derivation. Then the space ofprederivations of T ∗ G is Pder (cid:0) T ∗ G (cid:1) = der ( T ∗ G ) = span ( φ , φ , φ , φ , φ , φ , φ ) , where φ i , i = 1 , , , , , , are defined in Example 2.5.3. If φ is a prederivation of T ∗ G , φ = α φ + α φ + α φ + α φ + α φ + α φ + α φ , (4.123) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples where α i , i = 1 , , , , , , , are real numbers. Now, since G is simple then φ is skew-symmetric with respect to the form (4.118) if and only if it is an inner derivation (Propo-sition 4.5.1). Hence, α = 0 and then any µ a,b -skew-symmetric prederivation of T ∗ G hasthe form φ = α φ + α φ + α φ + α φ + α φ + α φ , (4.124)where α i , i = 1 , , , , , , are real numbers. In the basis of T ∗ G , such prederivation hasthe following matrix − α α − α − α α α α α α − α − α − α α α − α α − α − α (4.125) -dimensional Oscillator Lie Group In Example 3.5.3 we have defined the -dimensional osciallator Lie group G λ and its Liealgebra G λ . Proposition 4.6.4. Any bi-invariant metric on the -dimensional oscillator group isLorentzian and its induced orthogonal structure h , i λ on the Lie algebra G λ has the fol-lowing form : h x, y i λ = kµ λ ( x, y ) + mx − y − , (4.126) for any x = x − e − + x e + x e + ˇ x ˇ e and y = y − e − + y e + y e + ˇ y ˇ e in G λ , where µ λ is the orthogonal structure on G λ given by (3.78) and ( k, m ) is in R ∗ × R .Proof. We have already seen that the form µ λ given by (3.78) is an orthogonal structure on G λ . Then, any other orthogonal structure h , i λ on G λ is given by h x, y i λ = µ λ (cid:0) j ( x ) , y (cid:1) , for all x, y in G λ , where j : G λ → G λ in an µ λ -symmetric and invertible bi-invariant tensor. Now,one can check that such map j is given by j ( x ) = kx − e − +( mx − + kx ) e + kx e + k ˇ x ˇ e ,where ( k, m ) is in R ∗ × R . So, we have h x, y i λ = µ λ (cid:16) kx − e − + ( mx − + kx ) e + kx e + k ˇ x ˇ e , y − e − + y e + y e + ˇ y ˇ e (cid:17) = kx − y + ( mx − + kx ) y − + 1 λ ( kx y + k ˇ x ˇ y )= kx − y + mx − y − + kx y − + 1 λ ( kx y + k ˇ x ˇ y )= k h x − y + x y − + 1 λ ( x y + ˇ x ˇ y ) i + mx − y − = kµ λ ( x, y ) + mx − y − . Now the matrix of h , i λ on the basis ( e − , e , e , ˇ e ) of G λ is given by M λ = m k k kλ 00 0 0 kλ (4.127) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 xamples The characteristic polynomial of M λ is P λ ( t ) = (cid:0) t − kλ (cid:1) ( t − mt − k ) and its roots are t = kλ ; t = 12 (cid:0) m + √ m + 4 k (cid:1) ; t = 12 (cid:0) m − √ m + 4 k (cid:1) (4.128) t and t are simple roots while t is of multiplicity . It is clear that t ≥ , t ≤ and t has the same sign as k . Proposition 4.6.5. Any orthogonal structure µ ∗ λ on T ∗ G λ can be written as µ ∗ λ (cid:16) ( x, f ) , ( y, g ) (cid:17) = A h ( x, f ) , ( y, g ) i + Bµ λ ( x, y )+ C ( x − g + y − f )+ Dx − y − + Ef g , (4.129) for all elements ( x, f ) = x − e − + x e + x e + ˇ x ˇ e + f − e ∗− + f e ∗ + f e ∗ + ˇ f ˇ e ∗ and ( y, g ) = y − e − + y e + y e +ˇ y ˇ e + g − e ∗− + g e ∗ + g e ∗ +ˇ g ˇ e ∗ in T ∗ G λ , where ( A ; B, C, D, E ) is in R ∗ × R and h , i stands for the duality pairing between G λ and G ∗ λ .Proof. Since G λ is an orthogonal Lie algebra, then (see Theorem 4.4.1) any orthogonalstructure µ ∗ λ on T ∗ G λ is given by µ ∗ λ (cid:16) ( x, f ) , ( y, g ) (cid:17) = (cid:10) g , j ( x ) (cid:11) + (cid:10) f , j ( y ) (cid:11) + (cid:10) g, j ◦ θ − ( f ) (cid:11) + (cid:10) θ ◦ j ( x ) , y (cid:11) , for all ( x, f ) , ( y, g ) in T ∗ G λ , where θ : G λ → G ∗ λ , h θ ( x ) , y i = µ λ ( x, y ) for all x, y in G λ ; j , j and j commute with all adjoint operators of G and satisfy the following conditions :1. j is invertible ;2. j is µ λ -symmetric (cid:0) h , i λ being defined by (4.126) (cid:1) ;3. j ◦ ad x = 0 and h j ◦ θ − ( f ) , g i = h f, j ◦ θ − ( g ) i , for all x in G and all f, g in G ∗ .One can easily establish that the endomorphisms j , j and j have the following matriceson the basis ( e − , e , e , ˇ e ) of G λ : j = a a a a 00 0 0 a ; j = b b b b 00 0 0 b ; j = c where a = 0 , a , b , b , c are real numbers. It is also readily checked that, with respectto the basis ( e − , e , e , ˇ e ) and ( e − , e , e , ˇ e ) of G λ and G ∗ λ respectively, θ and θ − havethe following matrices : θ = λ 00 0 0 λ ; θ − = λ 00 0 0 λ (4.130)It is now a little matter to establish that µ ∗ λ (cid:16) ( x, f ) , ( y, g ) (cid:17) = a h ( x, f ) , ( y, g ) i + b µ λ ( x, y )+ a ( x − g + y − f ) + b x − y − + c f g . (4.131)This latter equality is nothing but (4.129); one just has to put A = a , B = b , C = a , D = b , E = c . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 hapter Five Kählerian Structure On The LieGroup of Affine Motions of theReal Line R Contents R . . . . . . . . . 112 An almost complex structure on a Lie algebra G , when it exists, is defined by a linearendomorphism J : G → G , J = − Id ( Id is the identity map of G ). If, in addition, J satisfies the condition J [ x, y ] − [ J x, y ] − [ x, J y ] − J [ J x, J y ] = 0 , for all x and y in G , we will say that J is integrable. Now, an integrable almost complexstructure is called a complex structure. In this case the pair ( G , J ) is called a complexalgebra.A metric µ on a complex algebra ( G , J ) is called Kählerian if it is hermitian, that is, µ ( J x, J y ) = µ ( x, y ) , for all vectors x and y in G , and if J is a parallel tensor with respect to the connectionarising from µ . Likewise, given a Lie algebra with metric µ , we shall say that a complexstructure J on G is Käahlerian if µ is Kählerian with respect to J in the above sense. Sucha pair ( J, µ ) defines a Kählerian structure on G .A Poisson-Lie structure on a Lie group G, is given by a Poisson tensor π on G, suchthat, when the Cartesian product G × G is equipped with the Poisson tensor π × π ,the multiplication m : ( σ, τ ) στ is a Poisson map between the Poisson manifolds On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ntroduction ( G × G, π × π ) and ( G, π ) . If f, g are in G ∗ and ¯ f , ¯ g are C ∞ functions on G with respectivederivatives f = ¯ f ∗ ,ǫ , g = ¯ g ∗ ,ǫ at the unit ǫ of G, one defines another element [ f, g ] ∗ of G ∗ by setting [ f, g ] ∗ := ( { ¯ f , ¯ g } ) ∗ ,ǫ . Then [ f, g ] ∗ does not depend on the choice of ¯ f and ¯ g as above, and ( G ∗ , [ , ] ∗ ) is a Liealgebra. Now, there is a symmetric role played by the spaces G and G ∗ , dual to each other.Indeed, as well as acting on G ∗ via the coadjoint action, G is also acted on by G ∗ using thecoadjoint action of ( G ∗ , [ , ] ∗ ) . A lot of the most interesting properties and applications of π, are encoded in the new Lie algebra ( G ⊕ G ∗ , [ , ] π ) , where [( x, f ) , ( y, g )] π := ([ x, y ] + ad ∗ f y − ad ∗ g x, ad ∗ x g − ad ∗ y f + [ f, g ] ∗ ) , (5.1)for every x, y in G and every f, g in G ∗ . The Lie algebras ( G ⊕ G ∗ , [ , ] π ) and ( G ∗ , [ , ] ∗ ) are respectively called the double andthe dual Lie algebras of the Poisson-Lie group ( G, π ) . Endowed with the duality pairingdefined in (2.2), the double Lie algebra of any Poisson-Lie group ( G, π ), is an orthogonalLie algebra, such that G and G ∗ are maximal totally isotropic (Lagrangian) subalgebras.Let r be an element of the wedge product ∧ G . Denote by r + (resp. r − ) the left (resp.right) invariant bivector field on G with value r = r + ǫ (resp. r = r − ǫ ) at ǫ . If π r := r + − r − isa Poisson tensor, then it is a Poisson-Lie tensor and r is called a solution of the Yang-BaxterEquation. If, in particular, r + is a (left invariant) Poisson tensor on G , then r is called asolution of the Classical Yang-Baxter Equation (CYBE) on G (or G ). In this latter case,the double Lie algebra ( G ⊕ G ∗ , [ , ] π r ) is isomorphic to the Lie algebra D of the cotangentbundle T ∗ G of G . See e.g. [28]. We may also consider the linear map ˜ r : G ∗ → G , where ˜ r ( f ) := r ( f, . ) . The linear map θ r : ( G ⊕ G ∗ , [ , ] π r ) → D , θ r ( x, f ) := ( x + ˜ r ( f ) , f ) , is an isomorphism of Lie algebras, between D and the double Lie algebra of any Poisson-Liegroup structure on G, given by a solution r of the CYBE.Let G = Aff ( R ) denote the group of affine motions of the real line R . It possesses a lotof interesting structures : symplectic ([1],[5]), complex, affine ([11]), Kälerian ([53]). Noteby G = aff ( R ) its Lie algebra. We wish to study the geometry of G as a Kählerian Liegroup. This supposes to describe the symplectic structures, the complex structures, theaffine transformations and transformations which preserve those structures. Furthermore,the symplectic structure corresponds to a solution of the Classical Yang-Baxter equation r (see [28]). So we will also study the double Lie group D ( G, r ) of G associated to r .In Section 5.2 we show how can man construct a sympectic structure on G , determinethe induced left-invariant affine structure, study the geodesics of this affine structure andcompare these geodesics with the integral curves of left-invariant vector fields on G . InSection 5.3 we deal with the geometry of a double Lie group of G associated to a solutionof the Classical Yang-Baxter equation. As explained below, the Lie algebra D ( G , r ) := G ⋉ G ∗ ( r ) of any double Lie group D ( G, r ) of G is isomorphic to the Lie algebra T ∗ G of thecotangent bundle T ∗ G := G ⋉ G ∗ ([28]). The Lie group structure on T ∗ G considered here is On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line the one obtained by semi-direct product via the coadjoint action of G on the dual space G ∗ of G , considered as an Abelian Lie group. We construct a double Lie group D ( G, r ) withLie algebra D ( G , r ) ≃ T ∗ G . The double Lie group D ( G, r ) admits an affine structure and acomplex structure ([28]). We study the both structures. R and its Lie algebra An affine transformation of R is a function R → R , x ax + b , where a and b are realnumbers ( a = 0 ). Let Aff ( R ) denote the set of all such transformations. We identify thesets Aff ( R ) and R ∗ × R by putting f = ( a, b ) if f : x ax + b . Now consider the operation ( a , b )( a , b ) := ( a a , a b + b ) , (5.2)for all ( a , b ) , ( a , b ) in Aff ( R ) . It is readily checked that, endowed with the compositionrule (5.2) Aff ( R ) is a group. The identity element is ǫ := (1 , and the inverse of theelement ( a, b ) ∈ G is given by ( a, b ) − = ( 1 a , − ba ) . Considering the underlying manifold R ∗ × R , G is a Lie group. We will note it by G . Remark 5.2.1. The operation law (5.2) is nothing but the composition law of maps. The Lie algebra of G is G = R (as a set). The bracket on G is given by [( u ′ , v ′ ) , ( u, v )] = (0 , − uv ′ + vu ′ ) . (5.3)In some basis ( e , e ) of G we have the following : [ e , e ] = e (5.4) We first recall the affine Lie group of R n ( n ∈ N ) and how can man construct a symplecticform on it (see [1]). The affine group of R n is the even dimensional Lie groupAff ( R n ) = (cid:26)(cid:18) A v (cid:19) , A ∈ GL ( n, R ) , v ∈ R n (cid:27) (5.5)Its Lie algebra is aff ( R n ) := (cid:26)(cid:18) A v (cid:19) , A ∈ gl ( n, R ) , v ∈ R n (cid:27) (5.6)Let e ij be the matrix such that the ( i − j ) -entry equals and the other entries are zero. ( e ij ) ≤ i ≤ n ≤ j ≤ n +1 forms a basis of aff ( n, R ) . Denote by ( e ∗ ij ) ≤ i ≤ n ≤ j ≤ n +1 its dual basis and set α := n X k =1 e ∗ k,k +1 and ω := − dα . (5.7) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line Hence, if x, y belong to aff ( n, R ) , ω ( x, y ) = − dα ( x, y ) , = α ([ x, y ]) One can check that the -form ω is non-degenerate and gives a left invariant symplecticstructure ω on Aff ( n, R ) by the following formula : ω g ( X | g , Y | g ) := ω ( T g L g − · X | g , T g L g − · Y | g ) , (5.8)for all g in Aff ( n, R ) and all vectors X | g , Y | g in T g Aff ( n, R ) , where L g : Aff ( n, R ) → Aff ( n, R ) , h g · h stands for the left translation by g in G .From now on we focus our attention on G = Aff ( R ) . The Lie algebra of G can bewritten G := aff ( R ) = { (cid:18) u v (cid:19) ; u, v ∈ R } (5.9)Put e := (cid:18) (cid:19) ; e := (cid:18) (cid:19) . Then G = vect { e , e } . We rename the vectors as follow : e = e , e = e . We have [ e , e ] = e . (5.10)Now we put α = e ∗ = e ∗ and ω = − dα = − de ∗ , that is ω ( x, y ) = e ∗ ([ x, y ]) , (5.11)for all x, y in G . We then have ω ( e , e ) = 0 ; ω ( e , e ) = 1 ; ω ( e , e ) = 0 . (5.12) ω is a non-degenerate -form on G . It induces a symplectic form ω on G by relation (5.8).For any ξ in G , let X ξ denote the associated left invariant vector field. That is X ξ | g := T ǫ L g · ξ, (5.13)for any g in G , where ǫ is the identity element of G . One defines an affine connection ∇ on G by the following formula (see [17]) : ∀ ξ, η, σ ∈ G , ω ( ∇ X ξ X η , X σ ) = − ω ( X η , [ X ξ , X σ ]) . (5.14)We obtain ∇ e e = − e ; ∇ e e = 0 ; ∇ e e = − e ; ∇ e e = 0 . (5.15)Note by Γ kij the symbols of Christoffel, that is ∇ e i e j = Γ kij e k (Einstein’s summation). Wethen have the following symbols of Christoffel Γ = − = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = − = 0 ; Γ = 0 (5.16)In the sequel, we will consider the connected component of the unit ǫ of G . Let us noteit by G . That is G = R ∗ + × R = { ( x, y ) ∈ R × R , x > } . We endow G with the connection(also denote by ∇ ) induced on G by the connection ∇ defined by relation (5.15). On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line ( G , ∇ ) at the unit Let ( ξ , η ) be an element of G and let γ = ( γ , γ ) be an auto-parallel ( ∇ ˙ γ ( t ) ˙ γ ( t ) = 0 , forall t ) curve such that γ (0) = ǫ = (1 , 0) ; ˙ γ (0) = ( ξ , η ) . (5.17)The curve γ satisfies the equations : ¨ γ − ˙ γ = 0 . (5.18) ¨ γ − ˙ γ ˙ γ = 0 . (5.19)Now we consider the following two cases. Case 1. γ (0) = (1 , and ˙ γ (0) = (0 , η ) .If γ is not constant, then for any t ∈ R such that γ ( t ) = 0 , we can solve equation (5.18)as follows : ¨ γ − ˙ γ = 0 ⇔ ¨ γ ˙ γ = 1 ⇔ − γ = t + c, c ∈ R ⇔ ˙ γ ( t ) = − t + c . (5.20)From the condition ˙ γ (0) = 0 , we obtain : − c = 0 , which is not possible, then ˙ γ ( t ) = 0 ,for all t ∈ R . We then have : • γ ( t ) = constante = a , for all real number t ; • Equation (5.18) gives γ ( t ) = bt + d , for all t in R , where b and d belong to R .From conditions γ (0) = (1 , and ˙ γ (0) = (0 , η ) , we have : a = 1 ; d = 0 ; b = η . (5.21)So, for any (0 , η ) in G the geodesic through ǫ with velocity (0 , η ) is given by γ ( t ) = (1 , η t ) , (5.22)for all t ∈ R . Case 2. Now we consider an element ( ξ , η ) of aff ( R ) , with ξ = 0 . Equation (5.18) canbe solve as above and ˙ γ ( t ) = − t + c . (5.23)From the condition ˙ γ (0) = ξ we obtain : − c = ξ , i.e. c = − ξ . Now we have : ˙ γ ( t ) = − t − ξ = − ξ ξ t − , t = 1 ξ γ ( t ) = − ln | ξ t − | + cste. (5.24) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line Since γ (0) = 1 , we have cste = 1 . It comes that : γ ( t ) = 1 − ln | ξ t − | , (5.25)for any t = 1 ξ and − eξ < t < eξ , since γ ( t ) > , if it is defined. Equation (5.19) nowbecomes ¨ γ + ξ ξ t − γ = 0 . (5.26)So we have ˙ γ ( t ) = C | ξ t − | , (5.27)where C is a real number. Since ˙ γ (0) = η , we have C = η and ˙ γ ( t ) = η | ξ t − | . (5.28)Integrating the latter we have : γ ( t ) = η ξ ln( k | ξ t − | ) , (5.29)where k > , t = 1 ξ and − eξ < t < eξ . Since γ (0) = 0 , we have k = 1 and γ ( t ) = η ξ ln | ξ t − | , (5.30)for any t = 1 ξ and − eξ < t < eξ . Hence, for any element ( ξ , η ) , with ξ = 0 , thegeodesic through ǫ with velocity ( ξ , η ) is given by : γ ( t ) = (cid:16) − ln | ξ t − | , η ξ ln | ξ t − | (cid:17) , (5.31)for all t = 1 ξ and − eξ < t < eξ .Now we can summarize. Proposition 5.2.1. A geodesic through the unit element of ( G , ∇ ) with velocity ( ξ, η ) in G is given by γ ( t ) = (1 , ηt ) if ξ = 0 , (1 − ln | ξt − | , ηξ ln | ξt − | ) if ξ = 0 . (5.32) for all t = 1 ξ and − eξ < t < eξ . In order to represent them, let us write Cartesian equations of the geodesics. Set γ ( t ) = ( y ( t ) , x ( t )) . (5.33) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line • If ξ = 0 , the Cartesian equation of the geodesic through ǫ with velocity (0 , η ) reads y = 1 . (5.34)This is the unique complete geodesic and is just a horizontal line through ǫ . • If ξ = 0 and η = 0 , the Cartesian equation of the geodesic through ǫ with velocity ( ξ, is given by (cid:26) x = 0 y > . (5.35)This geodesic is not complete and is the vertical line through ǫ contained in thehalf-plan { ( y, x ) ∈ R : y > } . • If ξ = 0 and η = 0 , the Cartesian equation of the geodesic through ǫ with velocity ( ξ, η ) is y = − ξη x + 1 and y > . We can simply write (cid:26) y = ax + 1 y > . a = R . (5.36)These geodesics are oblique lines through ǫ . They are not complete.The Figure 5.1 represents the geodesics of ( G , ∇ ) through ǫ .Figure 5.1: Geodesics of G through ǫ We just take the geodesics at t = 1 , whenever it is possible, to get the On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line The exponential map of the affine structure ∇ on G is defined for any ( ξ, η ) in G such that ξ = 1 and − e < ξ < e , by Exp ǫ ( ξ, η ) = (1 , η ) if ξ = 0 , (1 − ln | ξ − | , ηξ ln | ξ − | ) if ξ = 0 . (5.37)Let us now look at the inverse of Exp ǫ , whenever it exists. Let ( y, x ) be an element of G ǫ . We wish to find ( ξ, η ) in G such that Exp ǫ ( ξ, η ) = ( y, x ) .1. If y = 1 , then the unique solution is ( ξ, η ) = (0 , x ) .2. Suppose now y = 1 , then the equation Exp ǫ ( ξ, η ) = ( y, x ) gives the following twoequations : − ln | ξ − | = y (5.38) ηξ ln | ξ − | = x (5.39)Equation (5.38) is equivalent to ln | ξ − | = 1 − y | ξ − | = exp(1 − y ) ξ − (cid:26) exp(1 − y ) if ξ > − exp(1 − y ) if ξ < ξ = − y ) if ξ > − exp(1 − y ) if ξ < (5.40)Relation (5.39) becomes ηξ (1 − y ) = xη = ξ − y xη = − y )1 − y x if ξ > − exp(1 − y )1 − y x if ξ < (5.41)It comes that the logarithm map of ( G , ∇ ) which is the inverse of Exp ǫ , is defined onlyon the subset { (1 , x ) , x ∈ R } of G . Proposition 5.2.2. The Logarithm map of ( G , ∇ ) is defined on { (1 , x ) , x ∈ R } and isgiven by Log ǫ (1 , x ) = (0 , x ) . (5.42) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line G If g = ( y, x ) et h = ( y ′ , x ′ ) are two elements of G , the left translation L g by g acts on h as follows : L g h = ( yy ′ , yx ′ + x ) . (5.43)It comes that the differential map at ǫ of L g on the basis ( e , e ) has the following matrix : T ǫ L g = (cid:18) y y (cid:19) . Let ξ = ξ e + ξ e be an element of G . We wish to compute the exponential exp G ( ξ ) of ξ .Let X ξ stand for the left invariant vector field associated to ξ . We have : X ξ | g := T ǫ L g · ξ = (cid:18) y y (cid:19) (cid:18) ξ ξ (cid:19) = yξ ∂∂y + yξ ∂∂x . (5.44)Now let γ = ( γ , γ ) be the curve such that γ (0) = ǫ = (1 , and ˙ γ ( t ) = X ξ | γ ( t ) . (5.45)We then have the following equations : (cid:26) ˙ γ ( t ) = γ ( t ) ξ ;˙ γ ( t ) = γ ( t ) ξ . (5.46)The first relation of the system (5.46) has the following solution : γ ( t ) = k exp( ξ t ) , t ∈ R , (5.47)where k > . Since γ (0) = 1 , we have k = 1 and γ ( t ) = exp( ξ t ) , t ∈ R . (5.48)Now the second equation of the system (5.46) becomes : ˙ γ ( t ) = ξ exp( ξ t ) . (5.49)We consider two cases :(i) If ξ = 0 , then we have ˙ γ ( t ) = ξ γ ( t ) = ξ t + q, (5.50)where q is a real number. But γ (0) = 0 , then q = 0 and γ ( t ) = ξ t, t ∈ R . (5.51) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 he Affine Lie Group of the Real Line (ii) If ξ = 0 , then the equation (5.49) integrates to γ ( t ) = ξ ξ exp( ξ t ) + r, (5.52)for some real number r . Since, γ (0) = 0 , then ξ ξ + r = 0 , i.e. r = − ξ ξ . Hence, wehave : γ ( t ) = ξ ξ h exp( ξ t ) − i , (5.53)for all t in R .We then summarize in the Proposition 5.2.3. For any element ξ = ξ e + ξ e in G , the integral curve γ ξ of theleft-invariant vector field associated to ξ is defined by γ ( t ) = (1 , ξ t ) if ξ = 0; (cid:16) exp( ξ t ) , ξ ξ [exp( ξ t ) − (cid:17) if ξ = 0 . (5.54) for all t in R . It is readily checked that the Cartesian equation of these integral curves are given asfollows. Set γ ξ ( t ) = ( y, x ) . • If ξ = 0 , then the Cartesian equation of γ ξ is y = 1 . These curve is complete. • If ξ = 0 and ξ = 0 , then x = 0 and y > . This is a non-complete integral curve. • If ξ = 0 and ξ = 0 , we have the equation : y = ξ ξ x + 1 and y > or simply y = ax + 1 and y > , where a is a non-zero real number. Remark 5.2.2. The integral curves of the left-invariant vector fields associated to theelements of G globally coincides with the geodesics through ǫ of G obtained in (5.32). Corollary 5.2.2. The exponential map of the group G is defined by exp G ( ξ ) = (1 , ξ ) if ξ = 0; (cid:16) exp( ξ ) , ξ ξ [exp( ξ ) − (cid:17) if ξ = 0 . (5.55) for all ξ = ξ e + ξ e in G . Let us now define the Logarithm map of G . Let ( y, x ) ∈ G = R ∗ + × R . We want to find ξ ∈ G such that exp G ( ξ ) = ( y, x ) .1. If y = 1 , then the unique solution is ξ = (0 , x ) . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R 2. Suppose y = 1 , then (cid:16) exp( ξ ) , ξ ξ [exp( ξ ) − (cid:17) = ( y, x ) . (5.56)That is exp( ξ ) = yξ ξ [exp( ξ ) − 1] = x (5.57)and then ξ = ln yξ = xy − y. (5.58)We get, Proposition 5.2.4. The map exp G is invertible and its inverse is the map Log G : G → G defined by Log G ( y, x ) = (0 , x ) if y = 1 (cid:18) ln y , xy − y (cid:19) if y = 1 (5.59) R R Let G := Aff ( R ) be the affine Lie group of R and let G := aff ( R ) stand for the affine Liealgebra. It is shown (see [28]) that the double Lie algebra D ( G , r ) (where r : G ∗ → G isa solution of the Classical Yang-Baxter Equation), of a double Lie group D ( G, r ) of G , isisomorphic to the Lie algebra T ∗ G of the cotangent bundle T ∗ G = G ⋉ G ∗ endowed withthe Lie group structure obtained by semi-direct product via the coadjoint action of the Liegroup G and the Abelian Lie group G ∗ . The Lie bracket of T ∗ G , on some basis ( e , e , e , e ) ,reads : [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e . (5.60)We write T ∗ G = R e ⋉ H (see Example 2.5.1), where e acts on the Heisenberg Lie algebra H = span ( e , e , e ) by the restriction of ad e . Set D := ad e = diag (1 , , − . One Liegroup of T ∗ G is then the group R ⋉ H , where H is the -dimensional Heisenberg group( Lie ( H ) = H ) and R acts on H via the standard exponential of the endomorphism D by ρ : R → Aut ( H ) . Precisely, ρ t : y exp H (cid:16) Exp ( tD ) Y (cid:17) , (5.61)where exp H stands for the exponential map of the Lie group H , Exp is the standardexponential of endomorphisms and Y is an element of G such that exp H ( Y ) = y . Theproduct on R ⋉ H reads : ( t, x )( s, y ) = ( t + s, x · ρ t ( y )) , (5.62) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R where x · ρ t ( y ) is the product in the Heisenberg group of the elements x and ρ t ( y ) . We thenneed to know the exponential and the logarithm map of the Heisenberg group. Lemma 5.3.1. Let exp H and ln H be the exponential and the logarithm maps of the Heisen-berg group H . Then,1. the exponential map is defined by : for all ξ = ( ξ , ξ , ξ ) in H , exp H ( ξ ) = ( ξ , ξ + 12 ξ ξ , ξ ) (5.63) 2. while the logarithm is given by ln H ( y ) = ( y , y − y y , y ) , (5.64) for any ( y , y , y ) in H .Proof. Recall that the multiplication of H is given by : x · y = ( x , x , x ) · ( y , y , y ) = ( x + y , x + y + x y , x + y ) (5.65)and the unit element of H is ǫ H = (0 , , . The differential map at ǫ H of the left trans-lation by an element x has the following matrix on the basis ( e , e , e ) : T ǫ H L x = x . (5.66)Let us compute the exponential map exp H . Consider an element ξ = ξ e + ξ e + ξ e of H and let X ξ be the associated left invariant vector field. We note by γ ξ the integralcurve of X ξ with initial conditions γ ξ (0) = ǫ H and ˙ γ ξ (0) = ξ . We have : X ξ | ( x ,x ,x ) = T ǫ L ( x ,x ,x ) · ξ = ( ξ , ξ + x ξ , ξ ) (5.67)and ˙ γ ξ ( l ) = X ξ | γ ξ ( l ) . (5.68)If we set γ ξ = ( γ ξ , γ ξ , γ ξ ) , we obtain : ˙ γ ξ ( l ) = ξ (5.69) ˙ γ ξ ( l ) = ξ + γ ξ ( l ) ξ (5.70) ˙ γ ξ ( l ) = ξ . (5.71)With the initial conditions we obtain the following solution : γ ξ ( l ) = ξ l (5.72) γ ξ ( l ) = ξ l + 12 ξ ξ l (5.73) γ ξ ( l ) = ξ l. (5.74) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R It comes that exp H ( ξ ) = ( ξ , ξ + 12 ξ ξ , ξ ) . (5.75)Let now Y = ( Y , Y , Y ) be an element of H such that exp H ( Y ) = y . Then ( Y , Y + 12 Y Y , Y ) = ( y , y , y ) . We obtain that Y = y ; Y = y − y y ; Y = y . (5.76)That is ln H ( y ) = ( y , y − y y , y ) , (5.77)where ln H is the inverse map of exp H .Now we have : Exp ( tD ) = diag ( e t , , e − t ) = e t e − t . (5.78)Hence, ρ t ( y ) = exp H (cid:16) Exp ( tD ) Y (cid:17) = exp H (cid:16) y e t , y − y y , y e − t (cid:17) = (cid:16) y e t , y − y y + 12 y e t × y e − t , y e − t (cid:17) = (cid:16) y e t , y , y e − t (cid:17) (5.79)Now we can write the multplication on R ⋉ H as follows : ( t, x ) · ( s, y ) = ( t, x , x , x ) · ( s, y , y , y )= (cid:16) t + s , ( x , x , x ) · ρ t ( y , y , y ) (cid:17) = (cid:16) t + s , ( x , x , x ) · ( y e t , y , y e − t ) (cid:17) = (cid:16) t + s , x + y e t , x + y + x y e − t , x + y e − t (cid:17) . (5.80)Hence, the product of two elements x = ( x , x , x , x ) , y = ( y , y , y , y ) of R × H isgiven by x · y = (cid:16) x + y , x + y e x , x + y + x y e − x , x + y e − x (cid:17) . (5.81)The unit element is ǫ = (0 , , , and the inverse of an element x = ( x , x , x , x ) is theelement x − = ( − x , − x e − x , − x + x x , − x e x ) .We have prove the following Proposition 5.3.1. Endowed with the product (5.81), R × H is a double Lie group of theaffine Lie group of R . Let D ( G, r ) denote the double Lie group of G defined in Proposition 5.3.1. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R Among a lot of possibilities, we are interested in the left invariant affine connection on D ( G, r ) , introduced on any double Lie group of a symplectic Lie group by Diatta andMedina in [28] : ∇ ( x,α ) + ( y, β ) + = ( x · y + ad α y, ad ∗ r ( α ) β + ad ∗ x β ) + , (5.82)where x · y is the product induced by the symplectic structure on G through the formula ω ( x · y, z ) = − ω ( y, [ x, z ]) , (5.83)for all x, y, z ∈ G . Proposition 5.3.2. On the basis of D ( G , r ) , the connection is given by ∇ e e = − e ; ∇ e e = 0 ; ∇ e e = e ; ∇ e e = − e − e ∇ e e = − e ; ∇ e e = 0 ; ∇ e e = 0 ; ∇ e e = e ∇ e e = e ; ∇ e e = 0 ; ∇ e e = 0 ; ∇ e e = − e ∇ e e = − e − e ; ∇ e e = e ; ∇ e e = 0 ; ∇ e e = − e Proof. We have (see Relation (5.12)) ω ( e , e ) = 1 (5.84)and (see Relation (5.15)) e · e = − e ; e · e = − e . (5.85)Let us compute the bracket [ e , e ] ∗ on G ∗ ( r ) . [ e , e ] ∗ = ad ∗ r ( e ) e − ad ∗ r ( e ) e , (5.86)where r := q − : G ∗ G , with h q ( x ) , y i = ω ( x, y ) . h q ( e ) , e i = ω ( e , e ) = 0 . (5.87) h q ( e ) , e i = ω ( e , e ) = 1 . (5.88)Then, q ( e ) = e ∗ = e . h q ( e ) , e i = ω ( e , e ) = − . (5.89) h q ( e ) , e i = ω ( e , e ) = 0 . (5.90)Hence, q ( e ) = − e ∗ = − e . We then have the matrix of q on the basis ( e , e ) and ( e , e ) of G and G ∗ respectively : M = (cid:18) − 11 0 (cid:19) . (5.91)The matrix M est invertible and its inverse reads : M − = (cid:18) − (cid:19) . (5.92) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R It comes that r ( e ) = − e et r ( e ) = e . Hence, [ e , e ] ∗ = − ad ∗ e e − ad ∗ e e . = + e ◦ ad e + e ◦ ad e . (5.93) e ◦ ad e ( e i ) = (cid:26) − si i = 10 si i = 2 ⇒ e ◦ ad e = − e . (5.94) e ◦ ad e ( e i ) = (cid:26) si i = 10 si i = 2 ⇒ e ◦ ad e = 0 . (5.95)We then have [ e , e ] ∗ = − e . We can now compute the connection on the basis of D ( G , r ) . ∇ e e = e · e = − e . (5.96) ∇ e e = e · e = 0 . (5.97) ∇ e e = ad ∗ e e + ad ∗ e e = ad ∗ e e = − e ◦ ad e | G ∗ ( r ) = e (5.98) ∇ e e = ad ∗ e e + ad ∗ e e = − e ◦ ad e | G ∗ ( r ) + − e ◦ ad e | G = − e − e . (5.99) ∇ e e = e · e = − e . (5.100) ∇ e e = e · e = 0 . (5.101) ∇ e e = ad ∗ e e + ad ∗ e e = 0 . (5.102) ∇ e e = ad ∗ e e + ad ∗ e e = e . (5.103) ∇ e e = ∇ e e = e . (5.104) ∇ e e = ∇ e e = 0 . (5.105) ∇ e e = ad ∗ r ( e ) e = − ad ∗ e e = 0 . (5.106) ∇ e e = ad ∗ r ( e ) e = − ad ∗ e e = − e . (5.107) ∇ e e = ∇ e e = − e − e . (5.108) ∇ e e = ∇ e e = e . (5.109) ∇ e e = ad ∗ r ( e ) e = ad ∗ e e = 0 . ∇ e e = ad ∗ r ( e ) e = ad ∗ e e = − e . (5.110)Thus, if we set ∇ e i e j = Γ kij e k (Einstein’s summation), the Christoffel’s symbols Γ kij for On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R this connection are : Γ = − = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 1 ; Γ = 0 ; Γ = 0Γ = − = 0 ; Γ = 0 ; Γ = − = 0 ; Γ = − = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 1 ; Γ = 0Γ = 0 ; Γ = 1 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = − = 0Γ = − = 0 ; Γ = 0 ; Γ = − = 0 ; Γ = 0 ; Γ = 1 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 0 ; Γ = 0Γ = 0 ; Γ = 0 ; Γ = 0 ; Γ = − The only non vanishing symbols are : Γ = − = 1 ; Γ = − = − = − = 1 ; Γ = 1 ; Γ = − = − = − = 1 ; Γ = − ( D ( G, r ) , ∇ ) Now let γ ( t ) = (cid:0) γ ( t ) , γ ( t ) , γ ( t ) , γ ( t ) (cid:1) be a geodesic such that γ (0) = (0 , , , in T ∗ G ≃ D ( G, r ) and ˙ γ (0) = ( ξ , ξ , ξ , ξ ) ∈ D ( G , r ) . We have the following non-lineardifferential equations which are the equations of geodesics of D ( G, r ) . ¨ γ − ˙ γ − ˙ γ ˙ γ − ˙ γ ˙ γ = 0 , ¨ γ + ˙ γ ˙ γ − ˙ γ ˙ γ + ˙ γ ˙ γ = 0 , ¨ γ + ˙ γ ˙ γ − ˙ γ ˙ γ + ˙ γ ˙ γ = 0 , ¨ γ − ˙ γ − ˙ γ ˙ γ − ˙ γ ˙ γ = 0 . We rearrange the latter equations as follows : ¨ γ − ˙ γ − γ ˙ γ = 0 , (5.111) ¨ γ − ˙ γ ˙ γ + 2 ˙ γ ˙ γ = 0 , (5.112) ¨ γ − ˙ γ ˙ γ + 2 ˙ γ ˙ γ = 0 , (5.113) ¨ γ − ˙ γ − γ ˙ γ = 0 . (5.114)Unfortunately we do not yet have a solution for this system. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R R Let ξ = ξ e + ξ e + ξ e + ξ e be in D ( G , r ) . The left invariant vector field X ξ associatedto ξ is given by X ξ | ( x ,x ,x ,x ) = T ǫ L ( x ,x ,x ,x ) · ξ = e x x e − x e − x ξ ξ ξ ξ = ( ξ , ξ e x , ξ + x ξ e − x , ξ e − x ) . (5.115)Let γ ξ the unique curve such that γ ξ (0) = ǫ , ˙ γ ξ (0) = ξ and ˙ γ ξ ( t ) = X ξ | γ ξ ( t ) . (5.116)If γ ξ ( t ) = ( γ ξ ( t ) , γ ξ ( t ) , γ ξ ( t ) , γ ξ ( t )) , we have the following equations coming from (5.116) : ˙ γ ξ ( t ) = ξ (5.117) ˙ γ ξ ( t ) = ξ exp[ γ ξ ( t )] (5.118) ˙ γ ξ ( t ) = ξ + ξ γ ξ ( t ) exp[ − γ ξ ( t )] (5.119) ˙ γ ξ ( t ) = ξ exp[ − γ ξ ( t )] . (5.120)Resolving relation (5.117) and taking care with the initial conditions we have : γ ξ ( t ) = ξ t. (5.121)Relation (5.118) is resolved as follows. • If ξ = 0 , then γ ξ ( t ) = ξ t + b, b ∈ R = ξ t since γ ξ (0) = 0 . (5.122) • If ξ = 0 , then we have γ ξ ( t ) = ξ ξ exp( ξ t ) + b , b ∈ R ;= ξ ξ [exp( ξ t ) − , since γ ξ (0) = 0 . (5.123)Now we come to the relation (5.119) and again consider two cases. On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R • If ξ = 0 , the equation (5.119) can be written as ˙ γ ξ ( t ) = ξ + ξ ξ t. (5.124)The latter equation is solved as follows : γ ξ ( t ) = ξ t + 12 ξ ξ t + c , c ∈ R ;= ξ t + 12 ξ ξ t , as γ ξ (0) = 0 . (5.125) • If ξ = 0 , the relation (5.119) can be written as ˙ γ ξ ( t ) = ξ + ξ ξ ξ [exp( ξ t ) − 1] exp( − ξ t )= ξ + ξ ξ ξ [1 − exp( − ξ t )] . (5.126)and is integrated as γ ξ ( t ) = ξ t + ξ ξ ξ (cid:20) t + 1 ξ exp( − ξ t ) (cid:21) + c = ξ t + ξ ξ ξ (cid:20) t + 1 ξ exp( − ξ t ) (cid:21) − ξ ξ ξ , since γ ξ (0) = 0 . = (cid:18) ξ + ξ ξ ξ (cid:19) t + ξ ξ ξ [exp( − ξ t ) − . (5.127)Let us now solve equation (5.120). • If ξ = 0 , then (with the condition γ ξ (0) = 0 ) γ ξ ( t ) = ξ t. (5.128) • If ξ = 0 , then γ ξ ( t ) = − ξ ξ exp( − ξ t ) + d, d ∈ R ;= ξ ξ [1 − exp( − ξ t )] , since γ ξ (0) = 0 . (5.129)Let us summarize all the above. Proposition 5.3.3. The integral curve of the left invariant vector field associated to anyelement ξ = ξ e + ξ e + ξ e + ξ e of D ( G , r ) is defined by γ ξ ( t ) = (cid:18) , ξ t , ξ ξ t + ξ t , ξ t (cid:19) , (5.130) if ξ = 0 ; and by γ ξ ( t )= (cid:18) ξ t, ξ ξ [exp( ξ t ) − , (cid:18) ξ + ξ ξ ξ (cid:19) t + ξ ξ ξ [exp( − ξ t ) − , ξ ξ [1 − exp( − ξ t )] (cid:19) (5.131) if ξ = 0 . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R As an immediate consequence, we have the Corollary 5.3.1. The exponential map of the double Lie group D ( G, r ) of the affine Liegroup of R is defined as follows. For any ξ = ξ e + ξ e + ξ e + ξ e , exp D ( G,r ) ( ξ )= (cid:18) , ξ , ξ ξ + ξ , ξ (cid:19) , if ξ = 0 (cid:18) ξ , ξ ξ [exp( ξ ) − , ξ + ξ ξ ξ + ξ ξ ξ [exp( − ξ ) − , ξ ξ [1 − exp( − ξ )] (cid:19) if ξ = 0 . We are now going to deal with the invertibility of the exponential map above. Let ( x, y, z, t ) be an arbitrary element of D ( G, r ) . Our goal is to find ξ in T ∗ G such that exp D ( G,r ) ( ξ ) = ( x, y, z, t ) .1. If x = 0 , then (cid:18) , ξ , ξ ξ + ξ , ξ (cid:19) = ( x, y, z, t ) xξ = y ξ ξ + ξ = zξ = t ⇐⇒ xξ = yξ = z − ytξ = t (5.132)We then have that Lemma 5.3.2. The restriction exp |{ }× R of the exponential map of D ( G, r ) to thesubset { } × R of D ( G, r ) is invertible and its inverse is given by (0 , y, z, t ) (0 , y, z − yt, t ) (5.133)2. Suppose x = 0 , then we have (cid:18) ξ , ξ ξ [ e ξ − , ξ + ξ ξ ξ + ξ ξ ξ [ e − ξ − , ξ ξ [1 − e − ξ ] (cid:19) = ( x, y, z, t ) (5.134) ξ = xξ ξ [ e ξ − 1] = yξ + ξ ξ ξ + ξ ξ ξ [ e − ξ − 1] = zξ ξ [1 − e − ξ ] = t ⇔ ξ = xξ = xye x − ξ = z + xyt ( e x − e − x − 1) + yte x − ξ = xt − e − x Hence, On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R The restriction exp | R ∗ × R of the exponential map of the Lie group D ( G, r ) to the subset R ∗ × R is invertible and its inverse is the map defined as : ( x, y, z, t ) (cid:18) x, xye x − , z + yte x − h xe − x − i , xt − e − x (cid:19) (5.135)The precedent two Lemmas imply Proposition 5.3.4. The exponential map of the Lie group D ( G, r ) is invertible ans itsinverse is the map Log D ( G,r ) : D ( G, r ) → D ( G , r ) defined as follows : Log D ( G,r ) ( x, y, z, t ) = (0 , y, z − yt, t ) if x = 0 (cid:18) x, xye x − , z + yte x − h xe − x − i , xt − e − x (cid:19) if x = 0 (5.136) R From [28], the following formula defines a left-invariant complex structure J on any Liegroup with Lie algebra D ( G , r ) : J (cid:0) ( x, α ) + (cid:1) := (cid:0) − r ( α ) , q ( x ) (cid:1) + , (5.137)for any ( x, α ) in D ( G , r ) . We have J e +1 = ( q ( e )) + = e +4 = e x x e − x e − x = x e − x e − x (5.138) J e +2 = ( q ( e )) + = − e +3 = − e x x e − x e − x = − (5.139) J e +3 = ( − r ( e )) + = e +2 = e x x e − x e − x = e x (5.140) J e +4 = ( − r ( e )) + = − e +1 = − e x x e − x e − x = − (5.141) On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 ouble Lie groups of the affine Lie group of R Let us summarize J e +1 = x e − x ∂∂x + e − x ∂∂x (5.142) J e +2 = − ∂∂x (5.143) J e +3 = e x ∂∂x (5.144) J e +4 = − ∂∂x (5.145)This tensor is not bi-invariant, that is it does not commute with the adjoint operatorsof G . Indeed, if j := J | ǫ , we have • j [ e , e ] = je = − e ; • [ e , je ] = [ e , − e ] = 0 ,then j [ e , e ] = [ e , je ] . On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 eneral Conclusion In this thesis we study some aspect of the geometry of cotangent bundles of Lie groupsas Drinfel’d double Lie groups. Automorphisms of cotangent bundles of Lie groups arecompletely characterized and interesting results are obtained. We give prominence to thefact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super-symmetric Lie groups (Theorem 2.3.2). In the cases of semi-simple Lie algebras, compactLie algebras and more generally orthogonal Lie algebras, we recover by simple methodsinteresting co-homological known results (Section 2.3.6).Another theme in this thesis is the study of prederivations of cotangent bundles of Liegroups. The Lie algebra of prederivations encompasses the one of derivations as a subalge-bra. We find out that Lie algebras of cotangent Lie groups (which are not semi-simple) ofsemi-simple Lie groups have the property that all their prederivations are derivations. Thisresult is an extension of a well known result due to Müller ([64]). The structure of the Liealgebra of prederivations of Lie algebras of cotangent bundles of Lie groups is explore andwe have shown that the Lie algebra of prederivations of Lie algebras of cotangent bundleof Lie groups are reductive Lie algebras.Prederivations are useful tools for classifying objects like pseudo-Riemannian metrics([9], [64]). We have studied bi-invariant metrics on cotangent bundles of Lie groups andtheir isometries. The Lie algebra of the Lie group of isometries of a bi-invariant metricon a Lie group is entirely determine by prederivations of the Lie algebra which are skew-symmetric with respect to the induced orthogonal structure on the Lie algebra. We haveshown that the Lie group of isometries of any bi-invariant metric on the cotangent bundle ofany semi-simple Lie groups is given by inner derivations of the Lie algebra of the cotangentbundle.Last, we have dealt with an introduction to the geometry of the Lie group of affinemotions of the real line R , which is a Kählerian Lie group (see [53]). We describe, throughexplicit expressions, a symplectic structure, a complex structure, geodesics. Since the sym-plectic structure corresponds to a solution r of the Classical Yang-Baxter equation (see[28]), we also study the double Lie group associated to r .Admittedly, questions remain. Can it be otherwise ?Let G be a Lie algebra. We have said that the Lie algebra Pder ( T ∗ G ) of prederivationsof T ∗ G contains the one der ( T ∗ G ) of derivations as a Lie subalgebra. It would be interesting On the Geometry of Cotangent Bundles of Lie Groups Bakary MANGA c (cid:13) URMPM/IMSP 2010 eneral Conclusion to know : • if Pder ( T ∗ G ) can be decomposed into a semi-direct sum of der ( T ∗ G ) and a subspace h of Pder ( T ∗ G ) : Pder ( T ∗ G ) = der ( T ∗ G ) ⋉ h ; • if der ( T ∗ G ) is an ideal of Pder ( T ∗ G ) .We have seen that if G is semi-simple, then Pder ( T ∗ G ) = der ( T ∗ G ) . It would be a greatstep to find necessary and sufficient conditions for which the latter equality holds.Another open question is the following. As the cotangent bundle of any Lie group isa Drinfel’d double Lie group, it seems to be a good idea to extend the results within thisthesis to the class of Lie groups which are double Lie groups of Poisson-Lie groups.One other step is to list all orthogonal Lie groups of low-dimension using the doubleextension procedure of Medina and Revoy. 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