Generalized double extension and descriptions of qadratic Lie superalgebras
aa r X i v : . [ m a t h - ph ] D ec Generalized double extension and descriptions ofquadratic Lie superalgebras
Ignacio Bajo a, Sa¨ıd Benayadi b, Martin Bordemann ca Depto. Matem´atica Aplicada II, E.T.S.I. Telecomunicaci´on, Universidad de Vigo, CampusMarcosende, 36280 Vigo, Spain.
E-mail: [email protected] D´epartement de Math´ematique, Laboratoire LMAM, CNRS UMR 7122, Universit´e PaulVerlaine-Metz, Ile du saulcy, 57045 Metz cedex 1, France.
E-mail: [email protected] Laboratoire de Math´ematiques, Universit´e de Haute Alsace, 4, rue des Fr`eres Lumi`ere, 68093Mulhouse, France.
E-mail:
Abstract
A Lie superalgebra endowed with a supersymmetric, even, non-degenerate, invariant bilinearform is called a quadratic Lie superalgebra. In this paper we give an inductive descriptionof quadratic Lie superalgebras in terms of generalized double extensions; more precisely, weprove that every quadratic Lie superalgebra may be constructed by sucessive orthogonal sumsand/or generalized double extensions. In particular, we prove that all solvable quadratic Liesuperalgebras are obtained by a sequence of generalized double extensions by one-dimensionalLie superalgebras. We also prove that every solvable quadratic Lie superalgebra is isometricto either a T ∗ -extension of certain Lie superalgebra or to an ideal of codimension one of a T ∗ -extension. Keywords:
Simple Lie superalgebras, quadratic Lie superalgebra, double extension, T ∗ − extension,generalized double extension, cohomology of Lie superalgebras MSC:
Corresponding Author:
Sa¨ıd Benayadi, e-mail: [email protected] ntroduction
Let g = g ¯0 ⊕ g ¯1 a finite dimensional Lie superalgebra over a commutative algebraically closed field K of characteristic zero. An invariant scalar product B on g is an even, supersymmetric, non-degenerate bilinear form on g which is also invariant, this is to say, the equality B ([ X, Y ] , Z ) = B ( X, [ Y, Z ]) holds for every
X, Y, Z ∈ g . In this case, we say that ( g , B ) is a quadratic (ororthogonal or metrised) Lie superalgebra. The basic classical Lie superalgebras ([12], [19]) andsemisimple Lie algebras endowed with the Killing form are, therefore, examples of quadratic Liesuperalgebras, but there are many other Lie superalgebras which admit invariant scalar productseven when the Killing form is identically zero.Quadratic Lie algebras have been widely studied since they appear in connection with manyproblems derived from Geometry, Physics and other disciplines. For instance, it is well-knownthat the Lie algebra of a Lie group endowed with bi-invariant pseudo-Riemannian metric turnsout to be quadratic or –in terms of theoretical physics– that the Sugawara construction ispossible for these Lie algebras [9], which shows its relevance in Conformal Field Theory. In [16]Medina and Revoy introduced the notion of double extension which results as the combinationof a central extension and a semidirect product. This concept allowed them to give a certaininductive description of quadratic Lie algebras. More precisely, they proved that every quadraticLie algebra may be constructed as a direct sum of irreducible ones, and the latter by a sequence ofdouble extensions. A different construction, namely the T ∗ -extension, was given in [6] to describeall solvable quadratic Lie algebras (actually, the study in [6] is made not only in the case of Liealgebras, but for more general classes of algebras). The T ∗ -extension is essentially based ona generalized semidirect product. A different approach towards an inductive classification ofquadratic Lie algebras has been recently given by Kath and Olbrich in [14].The first attempt to use the technique of double extensions to describe quadratic Lie super-algebras was done in [3]. In this work the notion of double extension is defined for superalgebras,and it is shown that every irreducible quadratic Lie superalgebra in which the centre intersectsthe even part in a nontrivial way is a double extension. However, such a condition is ratherrestrictive and there do exist quadratic Lie superalgebras whose centre is contained in the oddpart. One can even find nilpotent examples (see section 4 below) in which such a phenomenonoccurs. Therefore, in order to give an inductive description of all quadratic Lie superalgebrasthere seems to be a need to generalize the concept of double extension. In this paper we shallintroduce the notion of generalized double extension which is at the same time a generalizationof the classical double extension and of the T ∗ -extension.The paper is organized in 5 sections. The first one just recalls the basic definitions andpreliminaries. In section 2 we introduce the concept of generalized double extension of quadraticLie superalgebras and show that classical double extensions and T ∗ -extensions may be seen asparticular cases. We then obtain a similar result to the one obtained by Medina and Revoy inthe Lie algebra case; this is done in the third section in which we explicitly prove that everyquadratic Lie algebra may be constructed by sucessive orthogonal sums and/or generalizeddouble extensions. The particular case of solvable Lie superalgebras is studied in sections 4 and5. We first see that every solvable quadratic Lie superalgebra may be obtained by a sequence ofgeneralized double extensions by one-dimensional superalgebra and then, in Section 5, we givea description of such solvable quadratic superalgebras via the T ∗ -extension. We show that, incontrast to the case of the classical double extension, the T ∗ -extension allows us to describe allsolvable cases. 1e are indebted to M. Duflo for providing the proof of the central Theorem 5.1, whichallowed us to extend a preliminary result concerning T ∗ -extensions which we had obtained (see[2]) to the general solvable case, and also for the second example of section 4. We also want toacknowledge the University of Vigo, the University of Metz, and the Graduiertenkolleg ‘PartielleDifferentialgleichungen’ of the University of Freiburg for the reserch stays during which the mainpart of this work was done. Notations : For a Z -graded vector space V over the field K we write V for its even part and V for its odd part, i.e. V = V ⊕ V . An element X of V is called homogeneous iff X ∈ V or X ∈ V . In this work, all elements are supposed to be homogeneous unless otherwise stated.For a homogeneous element X we shall use the standard notation | X | ∈ Z = { , } to indicateits degree, i.e. whether it is contained in the even part ( | X | = 0) or in the odd part ( | X | = 1).Moreover, for two integers k, l we denote their images in Z by k, l , and we use the well-definednotation ( − kl for ( − kl ∈ {− , } .All Lie superalgebras in this paper are finite-dimensional. Definition 1.1
Let g be a Lie superalgebra and let B be a bilinear form on g .i) B is called supersymmetric if B ( X, Y ) = ( − | X || Y | B ( Y, X ) , ∀ X ∈ g | X | , ∀ Y ∈ g | Y | .ii) B is called invariant if B ([ X, Y ] , Z ) = B ( X, [ Y, Z ]) , ∀ X, Y, Z ∈ g .iii) B is called even if B ( X, Y ) = 0 , ∀ X ∈ g ¯0 , ∀ Y ∈ g ¯1 . Definition 1.2 i) A Lie superalgebra g is called quadratic if there exists a bilinear form B on g such that B is supersymmetric, even, non-degenerate and invariant. In this case, B is called an invariant scalar product on g .ii) Let ( g , B ) be a quadratic Lie superalgebra.1) A graded ideal I of g is called non-degenerate (resp. degenerate) if the restriction of B to I × I is a non-degenerate (resp. degenerate) bilinear form.2) The quadratic Lie superalgebra ( g , B ) is called irreducible if g contains no non-degenerate graded ideal other than { } and I .3) We say that a graded ideal I of g is irreducible if I is non-degenerate and I containsno non-degenerate graded ideal of g other than { } and I .4) A graded ideal I of g will be said to be isotropic if B ( I , I ) = { } . Lemma 1.1
Let ( g , B ) be a quadratic Lie superalgebra. Let I be a graded ideal of g , we denoteby I ⊥ the orthogonal space of I with respect to B .i) I ⊥ is a graded ideal of g and [ I , I ⊥ ] = { } .ii) I is non-degenerate if and only if I ⊥ is non-degenerate.iii) If [ g , I ] = I , then I ⊥ = C g ( I ) , where C g ( I ) is the centralizer of I in g . Moreover, [ g , g ] ⊥ = z ( g ) . v) If I is non-degenerate, then ( I , ˜ B = B | I × I ) is a quadratic Lie superalgebra. Moreover thequadratic Lie superalgebra ( I , ˜ B ) is irreducible if and only if I is an irreducible graded idealof g .v) If H is a semisimple graded ideal of g , then H is non-degenerate and [ H , H ] = H . The following Proposition reduces the study of quadratic Lie superalgebras to those havingno non-degenerate graded ideal other than { } and I . Proposition 1.2
Let ( g , B ) be a quadratic Lie superalgebra. Then, g = ⊕ ni =1 g i , wherei) g i is a non-degenerate graded ideal, for all i ∈ { , . . . , n } ; ii) ( g i , B i = B | g i × g i ) is a quadratic Lie superalgebra, for all i ∈ { , . . . , n } ; iii) B ( g i , g j ) = { } , for all i, j ∈ { , . . . , n } . A proof of both, Lemma 1.1 and Proposition 1.2, can be found for instance in [5]. ¿From now on, if g is a Lie superalgebra, we will denote by Der( g ) the Lie superalgebra of allits superderivations and by End( g ) that of its linear endomorphisms. The symbol P cyclic willbe frequently used to denote cyclic sumation on a triple X, Y, Z.
Let ( g , [ , ] g ) and ( H , [ , ] H ) be two Lie superalgebras, F : g → Der( H ) be an even linear map (notnecessarily a morphism of Lie superalgebras), and L : g × g → H be an even superantisymmetricbilinear map such that the following equations are satisfied:[ F ( X ) , F ( Y )] − F ([ X, Y ] g ) = ad H L ( X, Y ) , (1) X cyclic ( − | X || Z | (cid:18) F ( X ) (cid:16) L ( X, Y ) (cid:17) − L ([ X, Y ] g , Z ) (cid:19) = 0 , ∀ ( X, Y, Z ) ∈ g | X | × g | Y | × g | Z | . (2)We define the following product [ , ] on the Z − graded K − vector space G = g ⊕ H :[ X + h, Y + l ] = [ X, Y ] g + F ( X ) (cid:16) l (cid:17) − ( − xy F ( Y ) (cid:16) h (cid:17) + L ( X, Y ) + [ h, l ] H , (3)for all ( X + h, Y + l ) ∈ G | X | × G | Y | .Using (1) and (2), we can see that the product [ , ] defined a Lie superalgebra structureon G . The Lie superalgebra G will be called the generalized semi-direct product of H by g bymeans of ( F, L ) (See [1] for more informations on this type of extension).
Remark 1
In particular, if L = 0 then G is the semi-direct product of H by g by means of F (cf. [19], Chapter III, 1). 3 .2 Generalized double extension of quadratic Lie superalgebras Let ( g , [ , ] , B ) a quadratic Lie superalgebra. We denote by Der a ( g ) the Lie superalgebra ofall superderivations of g which are superantisymmetric with respect to B . Let Out a ( g ) be thequotient of Der a ( g ) by all inner derivations of g which is known to be a Lie superalgebra. Let( g , [ , ] ) be an arbitrary Lie superalgebra of finite dimension and let B be a supersymmetricinvariant (not necessarily nondegenerate) bilinear form on g .Let φ : g → Der a ( g ) (4)be an even linear map, and ψ : g × g → g (5)be an even bilinear superantisymmetric map which satisfy the following twisted morphism con-dition for all homogeneous elements X, Y, Z ∈ g :[ φ ( X ) , φ ( Y )] − φ ([ X, Y ] ) = ad g ( ψ ( X, Y )) . (6)Note that this condition implies that the composed map ˜ φ : g φ → Der a ( g ) → Out a ( g ) is amorphism of Lie superalgebras. Lemma 2.1
With the above notation, the maps φ and ψ satisfy the following equation: X cyclic ( − | Z || X | ad g (cid:16) φ ( X )( ψ ( Y, Z )) − ψ ([ X, Y ] , Z ) (cid:17) = 0 . (7) Proof.
The Lemma easily follows by observing that the graded cyclic sum of [[ φ ( X ) , φ ( Y )] , φ ( Z )]vanishes by means of the super Jacobi identity and by using twice equation (6).This Lemma motivates the following, slightly stronger condition on φ and ψ : X cyclic ( − | Z || X | ( φ ( X )( ψ ( Y, Z )) − ψ ([ X, Y ] , Z )) = 0 . (8)Moreover, let χ : g × g → g ∗ (9)be the even bilinear map defined by the following equation for all homogeneous elements X ∈ g and A, B ∈ g : χ ( A, X )( B ) := − ( − | X || B | B ( ψ ( A, B ) , X ) . (10)Denoting the coadjoint representation of g on its dual by A.F := − (1) | A || F | F ◦ ad g ( A ) for allhomogeneous elements A ∈ g and F ∈ g ∗ , we finally need an even bilinear superantisymmetricmap w : g × g → g ∗ (11)satisfying a twisted cocycle condition X cyclic ( − | C || A | ( A.w ( B, C ) + w ( A, [ B, C ] ) + χ ( A, ψ ( B, C ))) = 0 . (12)and a supercyclicity condition w ( A, B )( C ) = ( − ( | B | + | C | ) | A | w ( B, C )( A ) , (13)4or all homogeneous elements A, B, C ∈ g . Let finallyΦ : g × g → g ∗ (14)be the even superantisymmetric bilinear map defined by the following equation for all homoge-neous elements X, Y ∈ g and A ∈ g :Φ( X, Y )( A ) := ( − | A | ( | X | + | Y | ) B ( φ ( A )( X ) , Y ) . (15)We shall subsume these notions in a Definition 2.1
Let ( g , B ) be a quadratic Lie superalgebra, g a Lie superalgebra and maps φ, ψ, w defined above satisfying the equations (6), (8), (12) and (13). We call ( g , B , g , φ, ψ, w )a context of generalized double extension (of the quadratic Lie superalgebra ( g , B ) by the Liesuperalgebra g .We have the following Lemma 2.2
Let ( g , B , g , φ, ψ, w ) be a context of generalized double extension of the quadraticLie superalgebra ( g , B ) by the Lie superalgebra g .Then the maps Φ and χ satisfy the following equations for all homogeneous elements X, Y ∈ g and A, B ∈ g : Φ( φ ( A ) X, Y ) + ( − | A || X | Φ( X, φ ( A ) Y ) − A. Φ( X, Y ) − χ ( A, [ X, Y ] ) = 0 , (16) and χ ([ A, B ] , X ) − χ ( A, φ ( B ) X ) + ( − | A || B | χ ( B, φ ( A ) X ) − A.χ ( B, X ) + ( − | A || B | B.χ ( A, X ) + Φ( ψ ( A, B ) , X ) = 0 (17) Moreover, Φ is an even two-cocycle of the Lie superalgebra g where g ∗ is a trivial module for g , i.e. for three homogeneous elements X, Y, Z ∈ g one has X cyclic ( − | X || Z | Φ( X, [ Y, Z ]) = 0 . (18) Proof.
Using the definitions of the maps φ, ψ, χ , and Φ the first two equations follow in astraight forward manner from the defining conditions (6) and (8), respectively. The cocyclecondition is a consequence of the fact that φ ( g ) is a subset of Der a ( g ) and has been shown in[3]. We are now in the position to put the structure of a quadratic Lie superalgebra on the vectorspace g := g ⊕ g ⊕ g ∗ : Theorem 2.3
Let ( g , B , g , φ, ψ, w ) be a context of generalized double extension of the quadraticLie superalgebra ( g , B ) by the Lie superalgebra g .Let A, B ∈ g , X, Y ∈ g , and F, H ∈ g ∗ homogeneous elements such that | A | = | X | = | F | and | B | = | Y | = | H | . We define the following bracket [ , ] on the vector space g : [ A + X + F, B + Y + H ] := [ A, B ] +[ X, Y ] + Φ( X, Y ) + ψ ( A, B ) + φ ( A )( Y ) − ( − | A || B | φ ( B )( X )+ A.H − ( − | A || B | B.F + χ ( A, Y ) − ( − | A || B | χ ( B, X )+ w ( A, B ) , (19)5 nd the following bilinear form B on g : B ( A + X + F, B + Y + H ) := B ( X, Y ) + F ( B ) + ( − | A || B | H ( A ) . (20) Then the triple ( g , [ , ] , B ) is a quadratic Lie superalgebra such that the subspace g ∗ is an isotropicideal of g and the subspace g ⊕ g ∗ is the orthogonal space of g ∗ .The quadratic Lie superalgebra g is called the generalized double extension of the quadraticLie superalgebra ( g , B ) by the Lie superalgebra g by means of ( φ, ψ, w ) . Proof.
By Lemma 2.2 we have Φ ∈ ( Z ( g , g ∗ )) ¯0 , it follows that we can define the followingLie superalgebra structure on the K − vector space g ⊕ g ∗ :[ X + F, Y + H ] = [ X, Y ] g + Φ( X, Y ) , ∀ X + F ∈ ( g ⊕ g ∗ ) | X | , Y + H ∈ ( g ⊕ g ∗ ) | Y | . This Lie superalgebra g ⊕ g ∗ is the central extension of g by g ∗ by means of Φ.Now, if A is an homogeneous element of g we consider ˜ φ ( A ) ∈ (End( g ⊕ g ∗ )) | A | defined by:˜ φ ( A )( X + F ):= φ ( A )( X )+ π ( A )( F )+ χ ( A, X ) , ∀ A + F ∈ g ⊕ g ∗ , (21)where π is the coadjoint representation of g . It is easy to see that equation (16) and the factthat φ ( A ) ∈ Der( g ) imply that ˜ φ ( A ) is a superderivation of Lie superalgebra g ⊕ g ∗ . It followsthat we have the even linear map ˜ φ : g → Der( g ⊕ g ∗ )defined by (21) on the homogeneous elements of g .Let us consider the even bilinear superantisymmetric map˜ ψ : g × g → g ⊕ g ∗ defined by: ˜ ψ ( A, B ) := ψ ( A, B ) + w ( A, B ) , ∀ A ∈ ( g ) | A | , B ∈ ( g ) | B | . By the equations (6), (17) we have[ ˜ φ ( A ) , ˜ φ ( B )] − ˜ φ ([ A, B ] ) = ad g ⊕ g ∗ ( ˜ ψ ( A, B )) , ∀ A ∈ ( g ) | A | , B ∈ ( g ) | B | . (22)Moreover, equations (8), (12) imply that X cyclic ( − | A || C | (cid:18) ˜ φ ( A ) (cid:16) ˜ ψ ( B, C ) (cid:17) − ˜ ψ ([ A, B ] g , C ) (cid:19) = 0 , ∀ ( A, B, C ) ∈ ( g ) | A | × ( g ) | B | × ( g ) | C | . (23)Consequently, equations (22), (23) imply that we can consider the generalized semi-direct prod-uct g := g ⊕ g ⊕ g ∗ of g ⊕ g ∗ by g by means of ( ˜ φ, ˜ ψ ). It is easy to verify that the bracket of g isexactly given by the bracket [ , ] of equation (19) defined in Theorem 2.3, and the bilinear form B as defined in (20) is an invariant scalar product on g . Finally, it is clear that the subspace g ∗ is an isotropic ideal of g , and that the subspace g ⊕ g ∗ is the orthogonal space of g ∗ . Remark 2 If g is the generalized double extension of the quadratic Lie superalgebra ( g , B )by the Lie superalgebra g by means of ( φ, ψ, w ) and if B is a supersymmetric invariant (notnecessarily nondegenerate) bilinear form on g , then the following bilinear form B ′ on g : B ′ ( A + X + F, C + Y + H ) := B ( A, C ) + B ( X, Y ) + F ( C ) + ( − | A || C | H ( A ) , (24)is another invariant scalar product on g . 6 .3 Double extension and T ∗ -extension: Important particular cases Two particular cases of the generalized double extension should be mentioned: double extensionand T ∗ -extensions. We first recall the notion of double extension of quadratic Lie superalgebrasas given in [3]: Theorem 2.4
Let ( g , B ) be a quadratic Lie superalgebra, g a Lie superalgebra and φ : g → Der a ( g ) ⊂ Der( g ) a morphism of Lie superalgebras.Let ψ be the map from g × g to g ∗ , defined by: ψ ( X, Y )( Z ) = ( − ( | X | + | Y | ) | Z | B ( ψ ( Z )( X ) , Y ) ∀ X ∈ ( g ) | X | , ∀ Y ∈ ( g ) | Y | , ∀ Z ∈ ( g ) | Z | . Let π be the coadjoint representation of g . Then the Z − graded vector space g = g ⊕ g ⊕ g ∗ with the product defined by [ X + X + f, Y + Y + g ] = [ X , Y ] g + [ X , Y ] g + φ ( X )( Y ) − ( − | X || Y | φ ( Y )( X )+ π ( X )( g ) − ( − | X || Y | π ( Y )( f ) + ψ ( X , Y ) , (where | X | = | X | = | X | = | f | and | Y | = | Y | = | Y | = | g | ) is a Lie superalgebra.Moreover, if γ is an invariant supersymmetric bilinear form on g , then the bilinear form T defined on g by T ( X + X + f, Y + Y + g ) = B ( X , Y ) + γ ( X , Y ) + f ( Y ) + ( − | X || Y | g ( X ) is an invariant scalar product on g .The Lie superalgebra g is called a double extension of ( g , B ) by g by means of φ . Let ( g , B , g , φ, ψ, w ) be a context of generalized double extension of the quadratic Liesuperalgebra ( g , B ) by the Lie superalgebra g . If ψ = 0 , w = 0 then the generalized doubleextension g of the quadratic Lie superalgebra ( g , B ) by the Lie superalgebra g by meansof ( φ, ψ, w ) is actually the double extension of ( g , B ) by g by means φ in the sense of theTheorem.Now, let ( g , B , g , φ, ψ = 0 , w ) be a context of generalized double extension of the quadraticLie superalgebra ( g = 0 , B = 0) by the Lie superalgebra g . Let g be the generalized doubleextension of the quadratic Lie superalgebra ( g = 0 , B = 0) by the Lie superalgebra g bymeans of ( φ = 0 , ψ = 0 , w ). The superalgebra g will be called the T ∗ -extension of g by meansof w , and we shall denote g by the symbol T ∗ w g or T ∗ g . In this case, by Theorem 2.3, we havethe following Proposition. Proposition 2.5
Let g be a Lie superalgebra, g ∗ its dual space, and π its coadjoint represen-tation. Let w be an even -cocycle of g with values in g ∗ . Define the following structures onthe vector space g := g ⊕ g ∗ : For a pair of homegeneous elements X + F, Y + H of g of degree | X | = | F | (resp . | Y | = | H | ), let [ X + F, Y + H ] g := [ X, Y ] g + w ( X, Y ) + π ( X )( H ) − ( − | X || Y | π ( Y )( F ) , and B ( X + F, Y + H ) := F ( Y ) + ( − | X || Y | H ( X ) . hen g endowed with the product [ , ] g is a Lie superalgebra.Moreover, ( g , B ) is a quadratic Lie superalgebra if and only if w is supercyclic, i.e. w ( X, Y )( Z ) = ( − | X | ( | Y | + | Z | ) w ( Y, Z )( X ) , ∀ ( X, Y, Z ) ∈ g | X | × g | Y | × g | Z | . We shall speak of the quadratic Lie superalgebra ( g , B ) constructed out of g and the supercyclic -cocycle w as a T ∗ -extension of g by means w and shall denote g by the symbol T ∗ w g or T ∗ g . Remark 3
It is clear from the definition that if g = T ∗ w ( g ), then its dimension n = dim( g )is even, and I = g ∗ is an isotropic graded ideal of dimension n/
2. Further, as a consequenceof Theorem 3.1 below, it is not difficult to see that the fact of being even-dimensional and theexistence of a graded ideal I such that I = I ⊥ characterize T ∗ -extensions.It should be noticed that if w is a supercyclic even 2-cocycle with values in g ∗ the the trilinearmapping ˆ w : ( g ) → K given by ˆ w ( X, Y, Z ) = w ( X, Y ) Z , turns out to be an even scalar 3-cocycle. Actually, one gets in this way an isomorphism between the subspace of ( Z ( g , g ∗ )) ¯0 composed of all supercyclic elements and ( Z ( g , K )) ¯0 . Therefore one easily gets the followingresult which gives a certain classification of the T ∗ -extensions of a given Lie algebra in terms ofthe cohomology class of ˆ w (see [2]). Proposition 2.6
Let g be a Lie superalgebra, w , w two supercyclic even 2-cocycles with valuesin g ∗ , and consider an even superantisymmetric bilinear mapping ϕ : g × g → K .The map S ϕ : T ∗ w ( g ) → T ∗ w ( g ) defined by S ϕ ( X + F ) = X + ϕ ( X, · ) + F defines an isometryof quadratic Lie superalgebras if and only if ˆ w − ˆ w is a coboundary. In this section, we are going to give an inductive description of quadratic Lie superalgebras byusing the notion of generalized double extension of quadratic Lie superalgebras.
Theorem 3.1
Let ( g , B ) be a quadratic Lie superalgebra, I be an isotropic graded ideal of g ,and I ⊥ its orthogonal space. Define1. the quadratic Lie superalgebra ( g , B ) with g := I ⊥ / I and B the quadratic form inducedby the restriction of B to I ⊥ × I ⊥ , and2. the Lie superalgebra g := g / I ⊥ .Then for any isotropic graded vector subspace V of g such that g = I ⊥ ⊕ V and any gradedsubspace A of I ⊥ with I ⊥ = A ⊕ I there is an even linear map φ : g → Der a ( g ) , an evenbilinear superantisymmetric map ψ : g × g → g , and an even bilinear superantisymmetricmap w : g × g → g ∗ such that ( g , B ) is isometric to the generalized double extension of thequadratic Lie superalgebra ( g , B ) by g by means of ( φ, ψ, w ) . Proof.
It is clear that I ⊆ I ⊥ since I is isotropic. Choose a graded vector subspace A of I ⊥ suchthat I ⊥ = A ⊕ I . Since I is the orthogonal space of I ⊥ it is obvious that A is nondegenerate.Hence A ⊥ is nondegenerate, and we have A ⊥ ∩ I ⊥ = I . Moreover, since A ∩ I = { } it follows8hat g = A ⊥ + I ⊥ Choose a graded vector subspace V of A ⊥ ⊆ g such that g = I ⊥ ⊕ V . Hence A ⊥ = V ⊕ I . Since the field K is algebraically closed and of characteristic different from 2 wemay choose the graded subspace V to be isotropic.For any A ∈ g = V ⊕ A ⊕ I denote by A V its component in V , by A A its component in A ,by A I its component in I , and by A I ⊥ its component in I ⊥ . Clearly, A = A V + A A + A I = A V + A I ⊥ . Moreover, let I ⊥ → g : X X be the natural projection which is a morphism ofLie superalgebras, and let K : V → g be the even linear map V ֒ → g → g which clearly is alinear isomorphism satisfying K ([ V , V ] V ) = [ KV , KV ] g for all V , V ∈ V . Using this notation we shall define the following three even maps φ : g → Hom K ( g , g ) : KV ( X [ V, X ]) ,ψ : g × g → g : ( KV , KV ) [ V , V ] I ⊥ ,w : g × g → g ∗ : ( KV , KV ) ( KV B ([ V , V ] , V )) , which are well-defined since K is a linear isomorphism and I ⊥ is a graded ideal of g .Since the ideals I ⊥ and I obviously are g -modules by means of the adjoint representation, itfollows that their quotient g is a g -module on which g acts by graded derivations. Thereforethe values of the linear map φ are graded derivations of g , a fact which also follows directly bythe graded Jacobi identity of g . Moreover since the induced quadratic form B on g is of theform B ( X, Y ) := B ( X, Y ) for all
X, Y ∈ I ⊥ we have for all V ∈ V B ( φ ( KV )( X ) , Y ) = B ([ V, X ] , Y ) = B ([ V, X ] , Y ) = − ( − | V || X | B ( X, [ V, Y ])= − ( − | V || X | B ( X, φ ( KV )( Y )) , whence φ takes its values in Der a ( g ) and satisfies the defining condition (4).Moreover, ψ is clearly an even superantisymmetric bilinear map. The fact that [ V , V ] =[ V , V ] V + [ V , V ] I ⊥ and the graded Jacobi identity on three elements V , V , V ∈ V show thatcondition (8) is satisfied on φ and ψ .Finally, it is obvious that w is an even superantisymmetric bilinear map. The supercycliccondition (13) for w follows from the invariance of B . Furthermore, for V , V , V , V ∈ V we getfor the map χ (see equation (9)): χ ( KV , ψ ( KV , KV ))( KV ) = − ( − ( | V | + | V | ) | V | B ( ψ ( KV , KV ) , ψ ( KV , KV ))= − ( − ( | V | + | V | ) | V | B ([ V , V ] I ⊥ , [ V , V ] I ⊥ )whence for one of the cyclic terms in condition (12) we get (cid:16) ( KV ) . ( w ( KV , KV )) (cid:17) ( KV ) + w ( KV , [ KV , KV ])( KV ) + χ ( KV , ψ ( KV , KV ))( KV ) = − ( − ( | V | + | V | ) | V | (cid:16) B ([ V , V ] , [ V , V ] V ) + B ([ V , V ] V , [ V , V ]) + B ([ V , V ] I ⊥ , [ V , V ] I ⊥ ) (cid:17) = − ( − ( | V | + | V | ) | V | (cid:16) B ([ V , V ] , [ V , V ]) (cid:17) since B ([ V , V ] V , [ V , V ]) = B ([ V , V ] V , [ V , V ] I ⊥ ) due to the fact that V is isotropic. Thegraded cyclic sum of the preceding terms vanishes, and thus condition (12) holds thanks to theinvariance of B and the graded Jacobi identity in g .9t follows that (cid:16) g , B , g , φ, ψ, w (cid:17) is a context of generalized double extension of the quadraticLie superalgebra ( g := I ⊥ / I , B ) by the Lie superalgebra g which is isomorphic to V as agraded vector space. Consider the generalized double extension ˜ g = g ⊕ g ⊕ g ∗ of ( g , B ) by g by means of ( φ, ψ, w ), we denote by ˜ B its invariant scalar product defined in Theorem 2.3.Let ∇ : I → g ∗ be the even linear map ∇ ( X )( KV ) := B ( X, V ) . It is easy to verify in a long, but straightforward manner that the following linear mapΠ : g = V ⊕ A ⊕ I → ˜ g = g ⊕ g ⊕ g ∗ defined by Π( V + A + X ) := KV + A + ∇ ( X )for all V ∈ V , A ∈ A , and X ∈ I is an isomorphism of Lie superalgebras. Moreover,˜ B (Π( X ) , Π( Y )) = B ( X, Y ) , ∀ X, Y ∈ g , i.e. Π is an isometry, which proves the Theorem. Corollary 3.2
Let ( g , B ) be an irreducible quadratic Lie superalgebra which is neither simplenor the one-dimensional Lie algebra. Then, ( g , B ) is a generalized double extension of a quadraticLie superalgebra ( A , T ) by a simple Lie superalgebra or by the one-dimensional Lie algebra orby the one-dimensional odd Lie superalgebra N = N ¯1 . Proof.
Since g is neither a simple nor the one-dimensional Lie algebra, then there exists anon-zero minimal graded ideal I of g . The fact that ( g , B ) is an irreducible quadratic Lie su-peralgebra implies that I is isotropic. Therefore, by Theorem 3.1, ( g , B ) is a generalized doubleextension of a quadratic Lie superalgebra ( I ⊥ / I , B ) by the Lie superalgebra g / I ⊥ . Since I isa minimal graded ideal of g , then I ⊥ is a maximal graded ideal of g . Consequently, g / I ⊥ iseither a simple Lie superalgebra or the one-dimensional Lie algebra or the one-dimensional oddLie superalgebra N = N ¯1 .Let B be the set consisting of { } , the isometry classes of quadratic simple Lie superalgebras,and the one-dimensional Lie algebra. Theorem 3.3
Let ( g , B ) be a quadratic Lie superalgebra. Then, either g is an element of B or ( g , B ) is obtained by a sequence of generalized double extensions by a simple Lie superalgebra orby the one-dimensional Lie algebra or by the one-dimensional Lie superalgebra N = N ¯1 and/ororthogonal direct sums of quadratic Lie superalgebras from a finite number of elements of B . Remark 4
It can be easily seen that the Cartan superalgebra W ( n ) = Der( V V ), where V is a n -dimensional vector space, is not quadratic for n ≥ B on W ( n ) the non-null subspace W n − = { D ∈ W ( n ) | D ( V ) ⊂ V n V } is orthogonal with respect to B to the whole W ( n ).In Proprosition 3 of [19, page 187] it is proved that if V is a n -dimensional vector space with n ≥ S ( V ) = ⊕ n − r = − S r ( V ) verifies [ S r ( V ) , S ( V )] = S r +1 ( V ) for all r ≥ − B is an even invariant bilinear supersymmetric form then B ( S n − ( V ) , S ( V )) = { } , which proves that B is degenerate and hence S ( V ) cannot be quadratic.The same reasoning shows that H ( n ) is not quadratic for n ≥ H r ( V ) , H ( V )] = H r +1 ( V ) for all r ≥ − H n − must beorthogonal to H ( n ) with respect to every even invariant bilinear supersymmetric form.We conclude that, with the possible exception of the superalebras ˜ S (2 r ), r ≥
1, a non-classical simple Lie superalgebra cannot be quadratic.
It has been proved in [3] that every n -dimensional irreducible quadratic Lie superalgebra g suchthat z ( g ) ∩ g ¯0 = { } is a double extension of a ( n − z ( g ) ∩ g ¯0 = { } is not always satisfied (evenin the nilpotent case) as the following examples show: Examples
1. For each positive integer n , let T ( n ) denote the linear space of square matrices of order n which are upper triangular and N ( n ) its subspace of strict upper triangular matrices. Itis easy to prove that g ( n ) = (cid:26)(cid:18) A BC D (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)
A, C, D ∈ N ( n ) , B ∈ T ( n ) (cid:27) is a Lie sub-superalgebra of gl ( n, n ) and that dim( z ( g ( n ))) = 1, z ( g ( n )) ⊂ g ( n ) ¯1 , and[ g ( n ) ¯1 , g ( n ) ¯1 ] = g ( n ) ¯0 . Let E n = T ∗ ( g ( n )) be the T ∗ − extension of g ( n ) by w = 0. It isclear that E n is nilpotent since so is g ( n ). Its centre is given by z ( E n ) = z ( g ( n )) ⊕ { f ∈ g ( n ) ∗ | f ([ g ( n ) , g ( n )]) = { }} and since [ g ( n ) ¯1 , g ( n ) ¯1 ] = g ( n ) ¯0 one easily gets that { } 6 = z ( E n ) ⊂ g ( n ) ¯1 ⊕ g ( n ) ∗ ¯1 = ( E n ) ¯1 .2. The superalgebras constructed in the example above have dimension 2 n (2 n −
1) and hence,the smallest one is the 12-dimensional superalgebra E . M. Duflo drew our attention tothe following 7-dimensional example: Consider the quadratic superspace V where theeven part is one-dimensional and the odd part has dimension 2 and let N be the standardnilpotent sub-superalgebra of osp (1 , V by N hasalso its centre contained in the odd part.Obviously, the results in [3] cannot be applied in these examples. Further, one can see thatthe second one gives a quadratic Lie superalgebra which cannot be constructed as a (classical)double extension by a one-dimensional algebra. Thus, the inductive classification of solvablequadratic Lie superalgebras requires the use of generalized double extensions. We will prove inthis section that, actually, every such superalgebra may be obtained by sucessive generalizeddouble extensions by one-dimensional superalgebras.11t is easy to verify that ( g , B , g := N , φ, ψ, w ) is a context of generalized double extensionof a quadratic Lie superalgebra ( g , B ) by the one-dimensional Lie superalgebra N = N ¯1 = K e ,if and only if w = 0 and there exists ( D, X ) ∈ [Der a ( g , B )] ¯1 × ( g ) ¯0 such that: φ ( e ) = D, ψ ( e, e ) = X , D ( X ) = 0 , B ( X , X ) = 0 and D = 12 ad g X . If g is the generalized double extension of the quadratic Lie superalgebra ( g , B ) by the Liesuperalgebra g := N by means of ( φ, ψ, w := 0), then the bracket [ , ] on g = K e ⊕ g ⊕ K e ∗ (where { e ∗ } is the dual basis of { e } ) is defined by:[ e, e ] = X , [ e ∗ , g ] = { } , [ e, X ] = D ( X ) − B ( X, X ) e ∗ , [ X, Y ] = [
X, Y ] g − B ( D ( X ) , Y ) e ∗ , ∀ X ∈ g | X | , Y ∈ g , and the invariant scalar product B on g is defined by: B ( e ∗ , e ) = 1 , B g × g := B ,B ( x, e ) = B ( x, e ∗ ) = B ( e ∗ , e ∗ ) = B ( e, e ) = 0 , ∀ X ∈ g . Proposition 4.1
Let ( g , B ) be an irreducible quadratic Lie superalgebra. If z ( g ) ∩ g ¯1 = { } ,then ( g , B ) is a generalized double extension of a quadratic Lie superalgebra ( A , T ) by the one-dimensional Lie superalgebra N = N ¯1 . Proof.
Suppose that z ( g ) ∩ g ¯1 = { } . Let X ∈ z ( g ) ∩ g ¯1 \{ } . Then I = K X is an isotropicgraded ideal of g . Therefore there exists Y ∈ g ¯1 such that B ( X, Y ) = 1, and B ( Y, Y ) = 0. Itfollows that g = I ⊥ ⊕ K Y . Then, by Theorem 3.1, ( g , B ) is a generalized double extension of aquadratic Lie superalgebra ( A , T ) by the one-dimensional Lie superalgebra N = N ¯1 . Corollary 4.2
Let ( g , B ) be an irreducible quadratic Lie superalgebra such that g is not the one-dimensional Lie algebra. If g is solvable, then either ( g , B ) is a double extension of a solvablequadratic Lie superalgebra ( A , T ) by the one-dimensional Lie algebra or ( g , B ) is a generalizeddouble extension of a solvable quadratic Lie superalgebra ( H , U ) by the one-dimensional Liesuperalgebra. Proof.
Since g is solvable, then z ( g ) = { } . If z ( g ) ∩ g ¯1 = { } , (resp. z ( g ) ∩ g ¯0 = { } ), then,by Corollary 4.1 (resp. by [3], Corollary 1 of Proposition 3.2.3), ( g , B ) is a generalized doubleextension of a quadratic Lie superalgebra ( H , U ) by the one-dimensional Lie superalgebra (resp.( g , B ) is a double extension of a quadratic Lie superalgebra ( A , T ) by the one-dimensional Liealgebra) where H = I ⊥ / I (resp. A = I ⊥ / I ) with I = K X where X ∈ z ( g ) ∩ g ¯1 \{ } (resp. X ∈ z ( g ) ∩ g ¯0 \{ } ). The fact that g is solvable implies that its graded ideal is also solvable, itfollows that the Lie superalgebra H (resp. A ) is solvable. Corollary 4.3
Let ( g , B ) be a solvable quadratic Lie superalgebra but neither { } nor the one-dimensional Lie algebra. Then, ( g , B ) is obtained by a sequence of double extensions of solvablequadratic Lie superalgebras by the one-dimensional Lie algebra and/or generalized double ex-tensions of solvable quadratic Lie superalgebras by the one-dimensional Lie superalgebra and/ororthogonal direct sums of solvable quadratic Lie superalgebras constructed in that way. Every Solvable Quadratic Lie Superalgebra is described by a T ∗ − extension The following central Theorem was communicated to us by M. Duflo. We have made its proofmore elementary upon circumventing the use of universal envelopping algebra which was usedin Duflo’s original proof.
Theorem 5.1 (Duflo)
Let g be a solvable Lie superalgebra and V be a finite-dimensional g -module. If V is simple, non-zero and isomorphic to its dual V ∗ , then V is the one-dimensionaltrivial g -module. Proof.
Let V be a simple module of g . We shall denote by ρ the module morphism g → Hom K ( V, V ), and by ρ ∗ the contragredient module map. . We shall first deal with the case that dim V = 1. It follows that V = K e . Let { f } bethe dual basis vector of { e } . Then there is a 1-form λ ∈ g ∗ such that for each homogeneouselement X of g we have ρ ( X ) e = λ ( X ) e and ρ ∗ ( X ) f = − ( − | X || e | λ ( X ) f . Let L : V ∗ → V be an isomorphism of g -modules. Then for each homogeneous element X of g the fact that L ( ρ ∗ ( X ) f ) = ρ ( X )( L ( f )) implies that (1 + ( − | X || e | ) λ ( X ) = 0. Consequently, λ ( X ) = 0 if | X | = 0. Since ρ ( X ) e = λ ( X ) e , it follows that λ ( X ) = 0 if | X | = 1 because the linear map ρ ( X )changes parity and e is homogeneous. We conclude that V is a trivial g -module.We are going to prove this Theorem now by induction on the dimension of g . We can and shallsuppose henceforth that dim V ≥ . If dim g = 0, then V is a simple K -vector space, whence dim V = 1 which contradicts theabove hypothesis. . If dim g = 1, then g = K X where X is a homogeneous element of g . Since K is algebraicallyclosed, it follows that ρ ( X ) has a nonzero eigenvector e ∈ V\{ } such that ρ ( X ) e = λe and λ ∈ K . Let e = e + e with e ∈ V and e ∈ V . In case X is even it follows that both e and e are eigenvectors of X for the eigenvalue λ . Since at least one of them is nonzero, therewould exist a proper submodule of V of dimension 1 which contradicts dim V ≥
2. In case X isodd, we have ρ ( X )( e ) = λe and ρ ( X )( e ) = λe . Therefore the subspace generated by e and e is a submodule of V , hence equal to it by simplicity of V . If e or e was zero, there wouldagain be a onedimensional submodule, contradiction. It follows that e ¯0 = 0 and e ¯1 = 0, whence V is 2-dimensional and spanned by e and e . Consider the dual basis { f ¯0 , f ¯1 } of V ∗ associatedto { e ¯0 , e ¯1 } . It follows that ρ ∗ ( X ) f ¯0 = − λf ¯1 and ρ ∗ ( X ) f ¯1 = λf ¯0 . Since V and its dual V ∗ areisomorphic g -modules, we can consider an isomorphism of g -modules L : V ∗ → V . Define matrixelements L a b ∈ K of L by L ( f b = L b e + L b e for all a, b ∈ Z . A little computation showsthat the isomorphism condition L ◦ ρ ∗ ( X ) = ρ ( X ) ◦ L implies that λ = 0. But then for instance K e would be one-dimensional submodule, contradiction. . Suppose now that dim g ≥
2, and the Theorem is true for all solvable Lie superalgebras H such that dim H < dim g . The fact that g is solvable implies that there exists a graded ideal h of g of codimension 1 such that [ g , g ] ⊂ h ⊂ g . There also exists a homogeneous element T ∈ g with g = h ⊕ K T (direct sum of vector spaces). Moreover, let W ⊂ V be a simple h -submoduleof V . Clearly, W is not equal to V since otherwise W = V would be one-dimensional thanks tothe induction hypothesis applied to h : this would contradict the assumption dim V ≥
2. Finally,for any nonnegative integer i let W i denote the following vector subspace of V : W i := W + ρ ( T ) W + · · · + ρ ( T ) i W . (25)13or strictly negative i we set W i := { } . We are going to show the following statements:There is a strictly positive integer M such that ρ ( T ) W M ⊂ W M and ρ ( T ) W M −
6⊂ W M − ; moreover if | T | = 1 then M = 1 . (26) W i is a h -module for all i ∈ N , and W M = V . (27)The h -module W i / W i − is isomorphic to W for all integers 0 ≤ i ≤ M (28)Indeed, to show statement (26), it is clear that ρ ( T ) W i ⊂ W i +1 , and since V is finite-dimensionalthere is a nonnegative integer N such that V , ρ ( T ) , . . . , ρ ( T ) N +1 are linearly dependent whichimplies that ρ ( T ) W N ⊂ W N . Then M will be the minimum of all these integers. The case M = 0 would imply that W = W is a g -module contradicting the simplicity of the g -module V whence M ≥
1. Note that for odd T we have ρ ([ T, T ]) = 2 ρ ( T ) with [ T, T ] ∈ [ g , g ] ⊂ h showing M = 1 since W is a h -module.Statement (27) follows for odd T immediately from the representation identity (for all homoge-neous x ∈ g ) ρ ( x ) ρ ( T ) = ρ ([ x, T ]) + ( − | T || x | ρ ( T ) ρ ( x ) (29)since M = 1 in that case, and W i = W M for all integers i ≥ M . For even T we use the followingidentity ρ ( x ) ρ ( T ) i = i X k =0 (cid:18) ik (cid:19) ρ ( T ) k ρ ([ . . . [ x, T ] . . . , T ] | {z } i − k brackets ) ∀ x ∈ g and ∀ i ∈ N , (30)(deduced from the representation identity (29) by induction) for the particular case x ∈ h toprove the statement since the iterated brackets [ . . . [ x, T ] . . . , T ] are then contained in the ideal h . Since the h -module W M is also stable by ρ ( T ) according to (26) it is a nonzero g -submoduleof the simple g -module V and therefore equal to V .For statement (28) consider the linear mapΦ i : W → W i / W i − : w ( − | T || w | ρ ( T ) i ( w ) + W i − . for homogeneous w ∈ W and all integers 0 ≤ i ≤ M . Since W i = ρ ( T ) i W + W i − it follows thatΦ i is surjective. For odd T it suffices to use the representation identity (29) to show that Φ i isa morphism of h -modules in view of the fact that M = 1, whereas for even T we use identity(30) to prove that each Φ i is a morphism of h -modules. In both cases the kernel of Φ i is an h -submodule of the simple h -module W so it is either equal to W or equal to { } : in the firstcase Φ i would be the zero map implying W i = W i − which is a contradiction to the definitionof the integer M . It follows that Φ i is injective and hence an isomorphism of h -modules.We continue the proof of the Theorem: statement (28) implies in particular for i = M thatthe h -module W is isomorphic to the quotient module V / W M − . Hence the dual h -module W ∗ is isomorphic to the h -module ( V / W M − ) ∗ which in turn is canonically isomorphic the the h -submodule W ann M − := { f ∈ V ∗ | f ( v ) = 0 ∀ v ∈ W M − } ⊂ V ∗ by means of the map ψ : W ann M − → ( V / W M − ) ∗ defined by ψ ( f )( v + W M − ) := f ( v ) (for all f ∈ W ann M − and v ∈ V ). Consider now an isomorphism of g -modules L : V ∗ → V , and let ˜ W bethe h -submodule of V defined by ˜ W := L ( W ann M − ). Since for each nonnegative integer i we have W i ⊂ W i +1 it follows that there is a smallest integer k such that ˜ W ⊂ W k and ˜ W 6⊂ W k − .14he image of ˜ W under the h -module map L ′ : W k → W k / W k − ∼ = W is an h -submodule of W , hence it is either equal to { } or to W . If the image was equal to { } , then ˜ W ⊂ W k − , acontradiction, and hence the image is equal to W . Sincedim ˜ W = dim( W ann M − ) = dim W ∗ = dim W it follows that ˜ W ∩ W k − = { } whence the composition of L ′ ◦ L restricted to W ann M − gives anisomorphism of the h -module W ∗ with the h -module W . By the induction hypothesis applied to h it follows that W is a one-dimensional trivial h -module. Again the representation identity (29)in case T is odd, and the identity (30) for even T show that then W M = V is a trivial h -module.Therefore V can in a canonical way be seen as a simple module of the one-dimensional quotientalgebra g / h satisfying the hypothesis of the Theorem. But this case has already been ruled outin . . Hence the hypothesis dim V ≥
Remark 5
The proof of Theorem 5.1 shows that this Theorem is still true in the case when K is algebraically closed with characteristic p = 2. Corollary 5.2
Let g be a solvable Lie superalgebra and V be a non-zero finite-dimensional g -module which admits a non-degenerate, even, supersymmetric and g -invariant bilinear form B .If dim V ≥ , then there exists a non-zero isotropic g -submodule W of V . Proof.
If dim
V ≥
2, then, by Theorem 5.1, V is a non-simple g -module. Consequently thereexists a simple g -submodule M of V , in particular M 6 = { } , and M 6 = V . Since M ⊥ ∩ M is a g -submodule of M , then M ⊥ ∩ M = { } or M ⊥ ∩ M = M . It follows that either B | M×M isnon-degenerate or
M ⊆ M ⊥ .If M ⊆ M ⊥ , then M is a non-zero isotropic g -submodule of V .If B | M×M is non-degenerate, then, by Theorem 5.1, dim M = 1 and M is a trivial g -module.Moreover, V = M ⊕ M ⊥ . Consider now a simple g -submodule M ′ of M ⊥ . If M ′ is isotropic,then the proof is finished. If M ′ is not isotropic, then B | M′×M′ is non-degenerate. It followsby Theorem 5.1, that M ′ is the one-dimensional trivial g -module. Consequently, the g -module M ⊕ M ′ is a 2-dimensional trivial nondegenerate g -module, and we can choose an isotropicone-dimensional graded vector sub-space W of M ⊕ M ′ . Therefore W is a non-zero isotropic g -submodule of V .The following Lemma is the Lie superalgebra analogue of Lemma 3.2 of [6]: Lemma 5.3
Let V be a non-zero finite dimensional Z -graded vector space which admits a non-degenerate, even and supersymmetric bilinear form B . Let L a solvable Lie sub-superalgebra of osp ( V , B ) and W a graded vector subspace of V .If W is an isotropic L -submodule of V , then there exists a graded vector subspace W max of V such that:(i) W ⊆ W max ; (ii) W max is an isotropic L -submodule of V ; (iii) W max is maximal among all isotropic graded vector sub-space of V ;15 iv) dim W max = [ n ] (i.e. the integer part of n );(v) If n is even, W max = ( W max ) ⊥ ; (vi) If n is odd, W max ⊆ ( W max ) ⊥ , dim( W max ) ⊥ − dim W max = 1 , and f (( W max ) ⊥ ) ⊆ W max ,for all f ∈ L . Proof.
We will prove this Lemma by induction on the dimension of V . It is clear that that thelemma is true if dim V = 1.Now, let us assume that the lemma is true if dim V < n , and we shall show it when dim V = n ,where n ≥
2. By the Corollary 5.2, there exists W a non-zero isotropic L -submodule of V . If W = W ⊥ , then dim W = n , it follows that W is maximal among all isotropic graded vectorsubspaces of V . Consequently, W max = W . If now W 6 = W ⊥ , we consider V ′ = W ⊥ / W which is a Z -graded vector space and dim V ′ < dim V = n Moreover, the well defined map¯ B : V ′ × V ′ → K defined by ¯ B ( X + W , Y + W ) := B ( X, Y ) , ∀ X, Y ∈ W ⊥ , is a non-degenerate,even and supersymmetric bilinear form. Now, if f ∈ L , then the linear map ¯ f : V ′ → V ′ definedby ¯ f ( X + W ) := f ( X ) + W , ∀ X ∈ W ⊥ , is well-defined. It is clear that L ′ = { ¯ f : f ∈ L} is asolvable Lie sub-superalgebra of osp ( V ′ , ¯ B ). It follows, by the induction hypothesis, that thereexists a graded vector subspace W ′ = { } max of V ′ which verifies the six assertions of the lemma.Let us consider W max := S − ( W ′ ), where S : W ⊥ → W ⊥ / W is the canonical surjection. Then, W max is a graded vector subspace of V such that W ⊆ W max and W ′ is isomorphic to W max / W .Consequently, dim W max = [ n ].Let X, Y ∈ W max . Since S ( X ) and S ( Y ) are elements of W ′ , then B ( X, Y ) = ¯ B ( S ( X ) , S ( Y )) =0. This proves that W max is isotropic. Consequently W max is maximal among all isotropicgraded vector subspaces of V because dim W max = [ n ]. Let X ∈ W max , f ∈ L , we have S ( f ( X )) = ¯ f ( S ( X )) ∈ W ′ then f ( X ) ∈ W max . It follows that W max is a L -submodule of V .If n is even it is clear that dim W max = dim( W max ) ⊥ = n . Consequently, W max = ( W max ) ⊥ .If n is odd, then there exists k ∈ N such that n = 2 k + 1 and dim W max = k . Con-sequently dim( W max ) ⊥ − dim W max = 1. Now, W ⊥ max / W max is an L -module and the map¯ B : W ⊥ max / W max × W ⊥ max / W max → K defined by ¯ B ( X + W max , Y + W max ) := B ( X, Y ) , ∀ X, Y ∈ ( W max ) ⊥ , is a L− invariant non-degenerate, even and supersymmetric bilinear form. Then, byTheorem 5.1, W ⊥ max / W max is a trivial L -module and, therefore, f (( W max ) ⊥ ) ⊆ W max , for all f ∈ L . Theorem 5.4
Let ( g , B ) be a solvable quadratic Lie superalgebra. Then g contains an isotropicgraded ideal I of dimension [ dim g ] which is maximal among all isotropic graded vector subspacesof g . Moreover, if dim g is even then ( g , B ) is isometric to some T ∗ − extension of the Liesuperalgebra g / I . If dim g is odd then ( g , B ) is isometric to a non-degenerate graded ideal ofcodimension one in some T ∗ − extension of the Lie superalgebra g / I . Proof.
The theorem is true if [ g , g ] = { } (i.e. g is abelian). Now, suppose that g is not abelian.Suppose que [ g , g ] = { } . Since g is solvable, then [ g , g ] = g and z ( g ) = { } because[ g , g ] ⊥ = z ( g ). If we suppose that the graded ideal [ g , g ] is non-degenerate then z ([ g , g ]) = { } which contradicts the fact that z ([ g , g ]) ⊆ [ g , g ] ∩ z ( g ) = { } . Consequently, [ g , g ] is degenerate.Therefore, z ( g ) ∩ [ g , g ] = { } . By Lemma 5.3, there exists an isotropic graded ideal I :=( z ( g ) ∩ [ g , g ]) max of g such that z ( g ) ∩ [ g , g ] ⊂ I , I is maximal among all isotropic gradedvector subspaces of g and dim I = [ dim g ]. If the dimension of g is even, then g is isometric16o a T ∗ − extension of g / I . Now, if the dimension of g is odd, it follows, by Lemma 5.3, thatdim I ⊥ − dim I = 1 and [ g , I ⊥ ] ⊆ I . Since B is non-degenerate and invariant, then [ I ⊥ , I ⊥ ] = { } .Since dim g ¯1 is even, then dim I ¯1 = dim g ¯1 , dim I ¯0 = [ dim g ¯0 ] and dim( I ¯0 ) ⊥ ∩ g ¯0 − dim I ¯0 = 1.It follows that ( I ¯0 ) ⊥ ∩ g ¯0 is not an isotropic vector subspace of g ¯0 . Consequently, there exists x ∈ ( I ¯0 ) ⊥ ∩ g ¯0 such that B ( x, x ) = 0. The fact that K is algebraically closed implies that thereexists α ∈ K such that B ( αx, αx ) = − E = K e with its invariant scalar product q defined by q ( e, e ) := 1, and the orthogonal direct sum ( A = g ⊕ E , T := B ⊥ q ) which isa quadratic Lie superalgebra such that A ¯0 = g ¯0 ⊕ E and A ¯1 = g ¯1 . Next we consider f := e + αX ∈ A ¯0 and H = I ⊕ K f . It is obvious that H is an isotropic graded ideal of A such thatdim H = dim A . It follows that A is isometric to a T ∗ − extension of the Lie superalgebra A / H .Let Φ : A → g / I be the linear map defined by: Φ( x + λe ) := ( x − λαX ) + I , ∀ ( x, λ ) ∈ g × K .It is a surjective morphism of Lie superalgebras such that KerΦ = H . Consequently, the Liesuperalgebras g / I and A / H are isomorphic. We conclude that A is isometric to a T ∗ − extensionof the Lie superalgebra g / I , and g is isometric to a non-degenerate graded ideal g ⊕ { } ofcodimension 1 of A . References [1] D.V. Alekseevsky, P.W. Michor and W. Ruppert, “Extensions of super Lie algebras”,
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