Integration and geometrization of Rota-Baxter Lie algebras
aa r X i v : . [ m a t h . QA ] S e p INTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS
LI GUO, HONGLEI LANG, AND YUNHE SHENGA bstract . This paper first introduces the notion of a Rota-Baxter operator (of weight 1) on a Liegroup so that its di ff erentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Di-rect products of Lie groups, including the decompositions of Iwasawa and Langlands, carry naturalRota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely thecrossed homomorphism on the Lie group, whose tangent map is the di ff erential operator of weight1 on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directlyon the Lie group level, the well-known global factorization theorems of Semenov-Tian-Shansky inhis study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids andRota-Baxter Lie groupoids are introduced, with the former a di ff erentiation of the latter. Further, aRota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-knownfact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of aRota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold.Examples and applications are provided for these new notions. C ontents
1. Introduction 21.1. Lie groupoids and Lie algebroids 21.2. Classical Yang-Baxter equations and Rota-Baxter Lie algebras 21.3. Rota-Baxter operators for other Lie structures 31.4. Outline of the paper 42. Rota-Baxter Lie groups and di ff erential Lie groups 42.1. Definition of Rota-Baxter Lie groups and examples 52.2. Rota-Baxter Lie groups as integrations of Rota-Baxter Lie algebras 72.3. Di ff erential operators on Lie groups and crossed homomorphisms 103. Factorization theorem of Rota-Baxter Lie groups 114. Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids 154.1. Rota-Baxter Lie algebroids 154.2. Relations to post-Lie algebroids 174.3. Rota-Baxter Lie groupoids and integration of Rota-Baxter Lie algebroids 20References 23 Date : October 1, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Rota-Baxter Lie group, Rota-Baxter Lie groupoid, Rota-Baxter Lie algebroid, post-Liealgebroid, integration.
1. I ntroduction
We define Rota-Baxter operators on Lie groups, Lie algebroids and Lie groupoids to give theintegration and geometrization of Rota-Baxter operators on Lie algebras. The relationship ofRota-Baxter Lie algebroids with post-Lie algebroids is established.1.1.
Lie groupoids and Lie algebroids.
Lie groups and Lie algebras are fundamental notions inmathematics. Their relationship, with a Lie algebra as the tangent of a Lie group, plays a majorrole in the study of geometry, algebra and mathematical physics. See for example [19, 21].Lie groupoids incorporate diverse objects, including Lie groups, manifolds, Lie group actionsand equivalence relations. They have played important roles in mathematical physics and non-commutative geometry. In particular, Lie groupoids unify internal and external symmetries andare used to describe singular quotient spaces in noncommutative geometry. The notion of a Liealgebroid was introduced by Pradines [32] in 1967, as a generalization of Lie algebras and tangentbundles. Just like Lie algebras being the infinitesimal objects of Lie groups, Lie algebroids arethe infinitesimal objects of Lie groupoids. Di ff erent from the case of Lie algebras and Lie groups,in general there is an obstruction class in integrating a Lie algebroid to a Lie groupoid [14]. Anatural class of Lie algebroids and Lie groupoids are given by the actions of Lie algebras and Liegroups on manifolds. See [27] for the general theory of Lie algebroids and Lie groupoids.These notions and operations can be summarized in the diagramLie algebras geometrization / / Lie algebroids Lie algebras action o o Lie groups geometrization / / di ff erentiation O O Lie groupoids di ff erentiation O O Lie groups. action o o di ff erentiation O O (1)1.2. Classical Yang-Baxter equations and Rota-Baxter Lie algebras.
The classical Yang-Baxter equation (CYBE) arose from the study of inverse scattering theory in the 1980s and thenwas recognized as the “semi-classical limit” of the quantum Yang-Baxter equation following theworks of C. N. Yang [42] and R. J. Baxter [5]. CYBE is further related to classical integrablesystems and quantum groups [12].An important method in studying the CYBE is the interpretation of it in various operator forms,beginning with the pioneering work of Semenov-Tian-Shansky [38] who showed that if there ex-ists a nondegenerate symmetric invariant bilinear form on a Lie algebra ( g , [ · , · ] g ) and if a solutionof the CYBE is skew-symmetric, then the solution can be equivalently expressed as a linear op-erator B : g → g satisfying the operator identity[ B ( u ) , B ( v )] g = B ([ B ( u ) , v ] g ) + B ([ u , B ( v )] g ) , ∀ u , v ∈ g . Also introduced in [38] is the modified Yang-Baxter equation (2) [ R ( u ) , R ( v )] g = R ([ R ( u ) , v ] g ) + R ([ u , R ( v )] g ) − [ u , v ] g , ∀ u , v ∈ g , from which the author obtained an Infinitesimal Factorization Theorem for the Lie algebra g and, after integration, his well-known Global Factorization Theorem for the corresponding Liegroup, with important applications to integrable systems [16, 20, 23, 24, 33, 34, 35]. Under thetransformation R = id + B , the operator R satisfies the modified Yang-Baxter equation if andonly if the operator B is the special case when λ = Rota-Baxter operator of weight λ ,characterized by the operator identity(3) [ B ( u ) , B ( v )] g = B ([ B ( u ) , v ] g ) + B ([ u , B ( v )] g ) + λ B ([ u , v ] g ) , ∀ u , v ∈ g , NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 3 for any scalar λ . A Lie algebra ( g , [ · , · ] g ) with a Rota-Baxter operator B of weight λ is called a Rota-Baxter Lie algebra of weight λ , denoted by ( g , [ · , · ] g , B ).As a remarkable coincidence, the associative analogs of Rota-Baxter operators had been intro-duced by G. Baxter in 1960 in his probability study [4] and pursued further by Atkinson, Cartierand foremost Rota from the perspectives of analysis and combinatorics [1, 11, 36, 37]. The re-markable renascence of the subject in this century has led to broad applications, especially in theConnes-Kreimer approach to renormalization of quantum field theory [13, 17].A Rota-Baxter Lie algebra with weight, and more generally an O -operator [6, 22], naturallygives rise to a pre-Lie algebra or a post-Lie algebra which has its origin in a study of operads [41]as a special case of the splitting of Lie algebras [2]. Pre-Lie algebras and post-Lie algebras playimportant roles in integrable systems and numerical integrations [3, 28]. See [7, 8, 9, 10] for moredetails about pre-Lie algebras and post-Lie algebras. Recently, the notion of post-Lie algebroidswas introduced and applied to numerical integrations [28, 29].1.3. Rota-Baxter operators for other Lie structures.
In light of the critical roles played byLie groups, Lie groupoids and Lie algebroids, as well as their relationship with Lie algebras asshown in the diagram (1) on the one hand, and Rota-Baxter Lie algebras and post-Lie algebrason the other, it is desirable to combine these notions and to extend the relationship among the Liestructures such as the one in (1). In particular, the fundamental Global Factorization Theorem ofSemenov-Tian-Shansky for a Lie group was obtained from integrating his Infinitesimal Factoriza-tion Theorem for a Lie algebra, making use of the modified Yang-Baxter equation (equivalently, aRota-Baxter operator of weight 1). Thus it is natural to ask whether is a Rota-Baxter operator onthe Lie group, so that the Global Factorization Theorem can be proved directly on the Lie grouplevel. These are the questions we address in this paper.The obvious challenge in giving the notion of a Rota-Baxter Lie group is that a Lie groupdoes not have the linear structure required for the original Rota-Baxter relation in (3). We usethe adjoint map to define an operator on a Lie group with a suitable operator identity (7) thathas properties similar to Rota-Baxter operators of weight 1 on Lie algebras, and such that thedi ff erentiation of such an operator on a Lie group is a Rota-Baxter operator of weight 1 on thecorresponding Lie algebra, thereby justifying naming it a Rota-Baxter operator (of weight 1) ona Lie group. The notion of a Rota-Baxter operator of weight − ff erential operator ofweight 1 on a Lie group since it is a formal inverse of the Rota-Baxter operator and its tangentmap is the usual di ff erential operator of weight 1 on a Lie algebra. We further show that the Rota-Baxter operator on a Lie group shares some important properties of the Rota-Baxter operatoron Lie algebra and give the aforementioned Global Factorization Theorem of Semenov-Tian-Shansky for Lie groups.We further generalize the notion of a Rota-Baxter operator to Lie algebroids by equipping aLie algebroid with a bundle map satisfying a Rota-Baxter type identity. The well-known fact thata Rota-Baxter operator on a Lie algebra gives rise to a post-Lie algebra finds its analog for Rota-Baxter operators on Lie algebroids. We show that the action Lie algebroid of the induced Liealgebra by a Rota-Baxter operator always admits an action post-Lie algebroid structure, whichcan be viewed as a particular case of [29, Theorem 4.4]. By geometrizing the notion of a Rota-Baxter operator on a Lie group, we obtain the notion of a Rota-Baxter operator on a Lie groupoidand show that the di ff erentiation of a Rota-Baxter operator on a Lie groupoid is a Rota-Baxteroperator on a Lie algebroid, as expected. LI GUO, HONGLEI LANG, AND YUNHE SHENG
We summarize the constructions and relations in diagram (4), enriching and extending dia-gram (1). The italic terms and dotted arrows are the ones introduced in this paper.
Lie algebras geometrization / / Lie algebroids actions ofLie algebras action o o post-Liealgebras geometrization / / subjacent O O post-Liealgebroids subjacent O O actions ofpost-Lie algebras action o o subjacent O O Rota-BaxterLie algebras geometrization / / splitting O O Rota-BaxterLie algebroids splitting O O actions of Rota-BaxterLie algebras action o o splitting O O Rota-BaxterLie groups descendent (cid:15) (cid:15) di ff erentiation O O geometrization / / Rota-BaxterLie groupoids descendent (cid:15) (cid:15) di ff erentiation O O actions of Rota-BaxterLie groups descendent (cid:15) (cid:15) action o o di ff erentiation O O Lie groups geometrization / / di ff erentiation @ @ Lie groupoids actions ofLie groups action o o di ff erentiation ] ] (4)1.4. Outline of the paper.
The paper is organized as follows. In Section 2, we first introducethe notion of a Rota-Baxter operator on a Lie group and show that projections to a direct factorLie group gives a Rota-Baxter operator, leading to examples from the Iwasawa decompositionand Langlands decomposition. We then justify this notion by showing that the di ff erentiation of aRota-Baxter operator on a Lie group is a Rota-Baxter operator on a Lie algebra (Theorem 2.10).A Rota-Baxter operator on a Lie group is also characterized by its graph, and taking the di ff er-entiation is also compatible with the derived product from a Rota-Baxter action. Di ff erential Liegroups are also introduced.In Section 3, we establish a factorization theorem for Rota-Baxter Lie groups (Theorem 3.5)which derives the factorization theorem of Semenov-Tian-Shansky for Lie groups.In Section 4, we introduce the geometric aspect into our study. First a Rota-Baxter Lie alge-broid is defined for a vector bundle, recovering a Rota-Baxter Lie algebra when the vector bundlereduces to a vector space. A Rota-Baxter Lie algebroid is shown to give a post-Lie algebroid in-troduced in [28] (Theorem 4.13). In particular, the notions of actions of Rota-Baxter Lie algebrasand of post-Lie algebras on manifolds are introduced to produce Rota-Baxter Lie algebroids andpost-Lie algebroids. Next we introduce the notion of a Rota-Baxter Lie groupoid and show thatit is the integration of a Rota-Baxter Lie algebroid (Theorem 4.25). The notion of an action of aRota-Baxter Lie group on a manifold is introduced, which gives rise to an action Rota-Baxter Liegroupoid and is compatible with the action Rota-Baxter Lie algebroid under di ff erentiation.2. R ota -B axter L ie groups and differential L ie groups In this section, we introduce the notions of Rota-Baxter Lie groups and di ff erential Lie groupsand show that they are the integrations of Rota-Baxter Lie algebras (of weight 1) and di ff erentialLie algebras (of weight 1) respectively. NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 5
Definition of Rota-Baxter Lie groups and examples.
We first give the notion of a Rota-Baxter Lie group and provide some examples.Let G be a group. For any g ∈ G , define the adjoint action Ad g : G −→ G , Ad g h : = ghg − , ∀ h ∈ G . The following two formulas will be frequently applied in the sequel:Ad g ( h h ) = Ad g h Ad g h , ∀ h , h ∈ G , (5) Ad g ◦ Ad g = Ad g g , ∀ g , g ∈ G . (6)Here is a general notion. Definition 2.1. A Rota-Baxter group is a group G with a map B : G −→ G such that B ( g ) B ( g ) = B ( g Ad B ( g ) g ) , ∀ g , g ∈ G , (7)called the Rota-Baxter relation for groups.When G is an abelian group, Eq. (7) states that B is a group homomorphism. We will mainlyconsider Lie groups in this paper. Definition 2.2.
Let G be a Lie group. A smooth map B : G −→ G satisfying relation (7) iscalled a Rota-Baxter operator on G . A Rota-Baxter Lie group is a Lie group equipped with aRota-Baxter operator.
Definition 2.3.
Let ( G , B ) and ( G ′ , B ′ ) be Rota-Baxter Lie groups. A smooth map Φ : G −→ G ′ is a Rota-Baxter Lie group homomorphism if Φ is a Lie group homomorphism such that Φ ◦ B = B ′ ◦ Φ . The scalar multiplication by − Example 2.4.
The inverse map ( · ) − : G −→ G is a Rota-Baxter operator on a Lie group G .Indeed the right hand side of Eq. (7) equals to(Ad g − g ) − g − = g − g − g g − = g − g − , which is exactly the left hand side of Eq. (7).For a Lie algebra g , if B : g → g is a Rota-Baxter operator of weight 1, then − id − B : g → g is also a Rota-Baxter operator of weight 1 on g . For a Lie group we have a similar result, notingthat − id − B = − id + B ( − id ). Proposition 2.5. If B : G → G is a Rota-Baxter operator on a Lie group G, then ˜ B : G → Gdefined by ˜ B ( g ) = g − B ( g − ) is also a Rota-Baxter operator on G.Proof. We show that ˜ B satisfies (7). In fact, for any g , h ∈ G ,˜ B ( g Ad ˜ B ( g ) h ) = ˜ B ( gg − B ( g − ) h B ( g − ) − g ) = g − B ( g − ) h − B ( g − ) − B ( g − B ( g − ) h − B ( g − ) − ) = g − B ( g − ) h − B ( g − ) − B ( g − ) B ( h − ) = ˜ B ( g ) ˜ B ( h ) , where for the second to the last equation we applied the Rota-Baxter relation (7) of B . (cid:3) LI GUO, HONGLEI LANG, AND YUNHE SHENG
The construction of solutions for the modified Yang-Baxter equation in [38] motivated thefollowing important class of Rota-Baxter Lie algebras.
Lemma 2.6.
Let g be an arbitrary Lie algebra and let g + and g − be its Lie subalgebras suchthat g = g + ⊕ g − as vector spaces. Denote by P + and P − the projections from g to g + and g − respectively. Then − P + and − P − are Rota-Baxter operators of weight .Proof. By the same argument as in the case of associative algebras [17], the projections P + and P − are Rota-Baxter operators of weight −
1. Thus − P + and − P − have weight 1. (cid:3) As an analog for Lie groups, we have
Lemma 2.7.
Let G be a Lie group and G + , G − be two Lie subgroups such that G = G + G − andG + ∩ G − = { e } . Define B : G → G by B ( g ) = g − − , ∀ g = g + g − , where g + ∈ G + , g − ∈ G − . Then ( G , B ) is a Rota-Baxter Lie group.Proof. Let g = g + g − and h = h + h − be two elements in G with g + , h + ∈ G + and g − , h − ∈ G − . Thenby the fact that G + and G − are Lie subgroups, we have B ( g Ad B ( g ) h ) = B ( g + g − g − − h + h − g − ) = B ( g + h + h − g − ) = ( h − g − ) − = B ( g ) B ( h ) . Hence B satisfies (7). (cid:3) Remark 2.8.
It is also natural to consider the inverse of the projection to the first factor. However,unlike the case of Lie algebras, the inverse of the projection to the first factor is not a Rota-Baxteroperator of weight 1. As we will see in Remark 2.12, the projection to the first factor is a Rota-Baxter operator of weight − Example 2.9. (i) By the Gram-Schmidt decomposition of matrices, we have the global de-composition of SL( n , C ), the space of complex matrices with determinant 1:SL( n , C ) = SU( n )SB( n , C ) , where SU( n ) is the space of unitary matrices with determinant 1 and SB( n , C ) consistsof all upper triangular matrices in SL( n , C ) with positive entries on the diagonal.Then (SL( n , C ) , B ) is a Rota-Baxter Lie group, where B ( ub ) = b − , for u ∈ SU( n ) and b ∈ SB( n , C ).More generally, for the Iwasawa decomposition [21, VI.4] G = KAN of a semisimplegroup G as the product of a compact subgroup, an abelian subgroup and a nilpotentsubgroup, the projection G → G , g = kan ( an ) − is a Rota-Baxter operator. Here notethat AN is a solvable Lie subgroup of G .(ii) The same conclusions can be obtained for some other decompositions of Lie groups,such as the Langlands decomposition [21, VII.7] P = MAN of a parabolic subgroup P of a reductive Lie group as the product of a semisimple subgroup, an abelian subgroupand a nilpotent subgroup. Note that AN is a Lie subgroup of P . NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 7
Rota-Baxter Lie groups as integrations of Rota-Baxter Lie algebras.
Generalizing thefact that Lie groups are integrations of Lie algebras, we now show that Rota-Baxter Lie groupsserve as integrations of Rota-Baxter Lie algebras of weight 1.Let G be a Lie group and e its identity. Let g = T e G be the Lie algebra of G and letexp ( · ) : g −→ G be the exponential map. Then the relation between the Lie bracket [ · , · ] g and the Lie group multi-plication is given by the following important formula:(8) [ u , v ] g = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = exp tu exp sv exp − tu , ∀ u , v ∈ g . Now we are ready to give the main result in this section, which states that the di ff erentiation ofa Rota-Baxter Lie group is a Rota-Baxter Lie algebra of weight 1. Theorem 2.10.
Let ( G , B ) be a Rota-Baxter Lie group. Let g = T e G be the Lie algebra of G and (9) B = B ∗ e : g −→ g the tangent map of B at the identity e. Then ( g , B ) is a Rota-Baxter Lie algebra of weight .Proof. Observe from (7) that B ( e ) = e . Since B = B ∗ e is the tangent map of B at e , we have thefollowing relation for su ffi ciently small t :(10) ddt (cid:12)(cid:12)(cid:12)(cid:12) t = B (exp tu ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = exp tB ( u ) = B ( u ) , ∀ u ∈ g . Now we check the identity[ B ( u ) , B ( v )] g = B ([ B ( u ) , v ] g + [ u , B ( v )] g + [ u , v ] g ) . By (5), (6), (8)-(10), and using the Leibniz rule, we obtain[ B ( u ) , B ( v )] g = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = exp tB ( u ) exp sB ( v ) exp − tB ( u ) ( Eq . (8)) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = B (exp tu ) B (exp sv ) B (exp − tu ) ( B = B ∗ e ) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = B (exp tu ) B (exp sv Ad B (exp sv ) exp − tu ) ( Eq . (7)) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = B (exp tu (Ad B (exp tu ) exp sv )(Ad B (exp tu ) B (exp sv ) exp − tu )) ( Eq . (7)) = B ∗ e d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = Ad B (exp tu ) exp sv + d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = Ad B (exp sv ) exp − tu + d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = exp tu exp sv exp − tu ! = B ([ B ( u ) , v ] g − [ B ( v ) , u ] g + [ u , v ] g ) . Therefore, ( g , B ) is a Rota-Baxter Lie algebra of weight 1. (cid:3) Example 2.11.
The tangent map of the inverse map ( · ) − : G −→ G is − id : g −→ g , whichis naturally a Rota-Baxter operator of weight 1 on the Lie algebra g . Moreover, the tangentmap of the Rota-Baxter operator B : G → G , B ( g + g − ) = g − − in Lemma 2.7 is − P − : g → g , − P − ( u + + u − ) = − u − in Lemma 2.6, which is a Rota-Baxter operator of weight 1 on g . LI GUO, HONGLEI LANG, AND YUNHE SHENG
Remark 2.12.
In the Lie algebra case, it is straightforward to see that ( g , B ) is a Rota-Baxter Liealgebra of weight 1 if and only if ( g , − B ) is a Rota-Baxter Lie algebra of weight −
1. Now at theLie group level, we can define a
Rota-Baxter Lie group of weight − G witha map C : G −→ G such that C ( g ) C ( g ) = C ((Ad C ( g ) g ) g ) , ∀ g , g ∈ G . (11)Let B be a Rota-Baxter operator on a Lie group G . Define C : G → G by C ( g ) : = B ( g − ) . In Eq. (7), replacing g and g by g − and g − , we obtain B ( g − ) B ( g − ) = B ( g − Ad B ( g − ) g − ) = B (cid:16) ( (cid:0) Ad B ( g − ) g (cid:1) g ) − (cid:17) . This gives C ( g ) C ( g ) = C (cid:16)(cid:0) Ad C ( g ) g (cid:1) g (cid:17) . Therefore, C is a Rota-Baxter operator of weight − C be the projection to the first direct factor, i.e. C ( g + g − ) : = g + . Then it is straightforward to deduce that C is a Rota-Baxter operator of weight − ff erentiation of a Rota-BaxterLie group of weight − −
1. Thus the above discussion canbe summarized into the following diagram:RB Lie alg. ( g , B ) of weight 1 additive inverse / / integration (cid:15) (cid:15) RB Lie alg. ( g , − B ) of weight − integration (cid:15) (cid:15) RB Lie group ( G , B ) of weight 1 multiplicative inverse / / RB Lie group ( G , C ) of weight − For simplicity, we will only consider Rota-Baxter Lie groups of weight in the rest of the paper. Now we give further characterization of Rota-Baxter operators on Lie groups.Let G be a Lie group. It acts on itself by the adjoint action Ad : G −→ Aut( G ). Thus we obtaina new Lie group structure on G × G given by( g , h ) · ( g , h ) : = ( g g , h Ad g h ) . We denote this Lie group by G ⊲ G .Let B : G −→ G be a smooth map and denote its graph in G × G by G B , that is, G B : = { ( B ( g ) , g ) | g ∈ G } . Proposition 2.13.
Let G be a Lie group. Then a smooth map B : G −→ G is a Rota-Baxteroperator if and only if the graph G B is a Lie subgroup of G ⊲ G.Proof.
For all g , h ∈ G , we have( B ( g ) , g ) · ( B ( h ) , h ) = ( B ( g ) B ( h ) , g Ad B ( g ) h ) , which implies that the multiplication is closed on the graph of B if and only if B ( g ) B ( h ) = B ( g Ad B ( g ) h ) , that is, B is a Rota-Baxter operator on the Lie group G . (cid:3) NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 9
For a Rota-Baxter Lie algebra ( g , B ) of weight 1, the bracket[ u , v ] B = [ B ( u ) , v ] g + [ u , B ( v )] g + [ u , v ] g , ∀ u , v ∈ g , (13)defines another Lie algebra structure on g , which we will call the descendent of [ · , · ] g , whilethe pair (( g , [ · , · ] g ) , ( g , [ · , · ] B ) is called the double in [38, 39]. Further B is also a Rota-Baxteroperator on ( g , [ · , · ] B ) and B : ( g , [ · , · ] B ) → g is a homomorphism of Rota-Baxter Lie algebras.This property also holds for Rota-Baxter Lie groups. Proposition 2.14.
Let ( G , B ) be a Rota-Baxter Lie group. (i) The pair ( G , ∗ ) , with the multiplicationg ∗ g : = g Ad B ( g ) g , ∀ g , g ∈ G , (14) is also a Lie group, called the descendent Lie group of the Rota-Baxter Lie group ( G , B ) ,whose Lie algebra is the descendent Lie algebra ( g , [ · , · ] B ) , where B = B ∗ e and [ · , · ] B isgiven in (13) . (ii) The operator B is a Rota-Baxter operator on the Lie group ( G , ∗ ) . (iii) The map B : ( G , ∗ ) → G is a homomorphism of Rota-Baxter Lie groups from ( G , ∗ , B ) to ( G , B ) .Proof. (i) Let e be the identity of the Lie group G . It is direct to see that B ( e ) = e . Also g ∗ e = e ∗ g = g and hence e is also the identity of the multiplication ∗ . Further,( g ∗ g ) ∗ g = ( g Ad B ( g ) g ) ∗ g = g Ad B ( g ) g Ad B ( g Ad B ( g g ) g and g ∗ ( g ∗ g ) = g Ad B ( g ) ( g Ad B ( g ) g ) = g Ad B ( g ) g Ad B ( g ) B ( g ) g . Hence the associativity holds since B satisfies (7).We can also see that the inverse of g for the multiplication ∗ , denoted by g † , is Ad B ( g ) − g − .By (8), we have[ u , v ] B = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = exp tu ∗ exp sv ∗ exp − tu = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = exp tu ∗ (exp sv Ad B (exp sv ) exp − tu ) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = exp tu (Ad B (exp tu ) exp sv )(Ad B (exp tu ) B (exp sv ) exp − tu ) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = Ad B (exp tu ) exp sv + d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = Ad B (exp sv ) exp − tu + d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = Ad exp tu exp sv = [ B ( u ) , v ] g − [ B ( v ) , u ] g + [ u , v ] g , which gives the desired identity.(ii) We shall check B ( g ) ∗ B ( h ) = B ( g ∗ B ( g ) ∗ h ∗ B ( g ) † ) , (15)where B ( g ) † is the inverse of B ( g ) with respect to the multiplication ∗ . By the fact that B ( g ∗ h ) = B ( g ) B ( h ) , we have e = B ( B ( g ) ∗ B ( g ) † ) = B ( g ) B ( B ( g ) † ) , which implies that B ( g ) − = B ( B ( g ) † ). Based on this and the definition of ∗ , we have B ( g ) ∗ B ( h ) = B ( g ) B ( g ) B ( h ) B ( g ) − = B ( g ∗ B ( g ) ∗ h ∗ B ( g ) † ) . This is (15).(iii) It follows from the definition of Rota-Baxter Lie groups that B is a homomorphism of Liegroups. Further the compatibility of B with the Rota-Baxter operators is tautology: B ◦ B = B ◦ B . (cid:3) Corollary 2.15.
For the Rota-Baxter operator B : G → G given in Lemma 2.7, we have a newgroup structure on G given byg ∗ h = ( g + g − ) ∗ ( h + h − ) = g + g − g − − h + h − g − = g + h + h − g − , ∀ g , h ∈ G . The Lie algebra of this Lie group is given by [ u , v ] B = [ u + + u − , v + + v − ] B = − [ u − , v + + v − ] g − [ u + + u − , v − ] g + [ u + + u − , v + + v − ] g = [ u + , v + ] g − [ u − , v − ] g , which corresponds to the Rota-Baxter operator of weight given in Lemma 2.6. Di ff erential operators on Lie groups and crossed homomorphisms. Recall that a di ff er-ential operator (of weight λ ) on a Lie algebra is a linear map D : g → g such that D [ u , v ] g = [ D ( u ) , v ] g + [ u , D ( v )] g + λ [ D ( u ) , D ( v )] g . As in the case for associative algebras [18], the di ff erential operator of weight λ on a Lie algebrais the formal left inverse of the Rota-Baxter operator of weight λ on a Lie algebra. It is thusnatural to look for a di ff erential operator of weight ± ff erential operator on Liegroups should be the formal inverse of the Rota-Baxter operator on Lie groups;(ii) The tangent map of the di ff erential operator on Lie groups should be the di ff erentialoperator on Lie algebras.We now show that these requirements are fulfilled by the notion of a crossed homomorphismor a 1-cocycle with non-abelian coe ffi cients [40]. Definition 2.16.
Let G be a group and let Γ be a group with a group action by G , given by α α x , α ∈ Γ , x ∈ G . A map f : G → Γ is called a crossed homomorphism or a if f ( xy ) = f ( x ) f ( y ) x , ∀ x , y ∈ G . Taking Γ to be the group G itself equipped with the conjugation action of G , we give thefollowing notion. Definition 2.17.
A smooth map D on a Lie group G is called a di ff erential operator of weight −
1) if(16) D ( gh ) = D ( g )Ad g D ( h ) (cid:0) resp. D ( gh ) = (Ad g D ( h )) D ( g ) (cid:1) , ∀ g , h ∈ G . Then ( G , D ) is called a di ff erential Lie group of weight − Theorem 2.18. (i)
The formal inverse of a Rota-Baxter operator of weight ± on Lie groupsis the di ff erential operator of ± . NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 11 (ii)
Let ( G , D ) be a di ff erential Lie group of weight ± . Let g be the Lie algebra of the Liegroup G and let D = D ∗ e be the tangent map of D . Then ( g , D ) is a di ff erential Liealgebra of the same weight.Proof. (i) Let D be a formal inverse of the Rota-Baxter operator B on a Lie group G : B ( g ) B ( h ) = B ( g Ad B ( g ) h ) , ∀ g , h ∈ G . Replacing B ( g ) and B ( h ) by x and y respectively leads to replacing g and h by D ( x ) and D ( y )respectively. Then applying D to both sides of the above equation gives D ( xy ) = D ( x )Ad x D ( y ) , ∀ x , y ∈ G . Thus D is a di ff erential operator of weight 1. The same proof works for the case of weight − D : G → G be a di ff erential operator of weight 1. Note that D ( e ) = e . We claim that thetangent map of D at the identity D = D ∗ e : g → g satisfies D [ u , v ] g = [ D ( u ) , v ] g + [ u , D ( v )] g + [ D ( u ) , D ( v )] g . In fact, by ddt (cid:12)(cid:12)(cid:12)(cid:12) t = D (exp tu ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = exp tD ( u ) = D ( u ), (8) and (16), we have D [ u , v ] g = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = D (exp tu exp sv exp − tu ) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = D (exp tu exp sv )Ad exp tu exp sv D (exp − tu ) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = D (exp tu )Ad exp tu D (exp sv )Ad exp tu exp sv D (exp − tu ) = d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = Ad exp sv D (exp − tu ) + d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = Ad exp tu D (exp sv ) + d dtds (cid:12)(cid:12)(cid:12)(cid:12) t , s = D (exp tu ) D (exp sv ) D (exp − tu ) = [ D ( u ) , v ] g + [ u , D ( v )] g + [ D ( u ) , D ( v )] g . The same proof applies to the case when the weight is − (cid:3)
3. F actorization theorem of R ota -B axter L ie groups In this section, we prove a factorization theorem of Rota-Baxter Lie groups which yields theGlobal Factorization Theorem of Semenov-Tian-Shansky for Lie groups given in [38, 39]. Thenotion of Rota-Baxter operators on Lie groups allows us to adapt the approach in [38, 39] fromLie algebras to Lie groups. See [16] for a similar factorization on Poisson Lie groups.Let ( G , B ) be a Rota-Baxter Lie group. Denote by G B the Lie group G with the new groupstructure g ∗ g : = g Ad B ( g ) g . Proposition 3.1.
Let ( G , B ) be a Rota-Baxter Lie group. Define B + : G → G , B + ( g ) = g B ( g ) . Then B + is a Lie group homomorphism from G B to G. Proof.
We have B + ( g Ad B ( g ) g ) = g (Ad B ( g ) g ) B ( g Ad B ( g ) g ) = g B ( g ) g B ( g ) − B ( g ) B ( g ) = g B ( g ) g B ( g ) = B + ( g ) B + ( g ) . So B + is a Lie group homomorphism. (cid:3) Let B be a Rota-Baxter operator on G . Define four subsets of G as follows: G + : = Im B + , G − : = Im B , K + : = ker B , K − : = ker B + . Since both B and B + are Lie group homomorphisms, G + and G − are Lie subgroups of G , K + and K − are normal Lie subgroups of G B , and G ± (cid:27) G B / K ∓ . Moreover, we have the followingrelations. Lemma 3.2. K + ⊂ G + and K − ⊂ G − are normal Lie subgroups.Proof. Let k ∈ K − , that is, B + ( k ) = k B ( k ) = e . Then we have k = B ( k ) − = B ( k † ) by (7), where k † is the inverse of k in G B . Thus k ∈ G − and K − ⊂ G − .For any k ∈ K − and B ( g ) ∈ G − , let us check B ( g ) k B ( g ) − ∈ K − . If fact, note that g † = Ad B ( g ) − g − . Then we have g ∗ k ∗ g † = g Ad B ( g ) k Ad B ( g ) B ( k ) Ad B ( g ) − g − = g B ( g ) k B ( g ) − B ( g ) B ( k ) B ( g ) − g − B ( g ) B ( k ) − B ( g ) − = B ( g ) k B ( g ) − , where in the last equation we used the fact that k B ( k ) = e . Thus, B ( g ) k B ( g ) − B ( B ( g ) k B ( g ) − ) = B ( g ) k B ( g ) − B ( g ∗ k ∗ g † ) = B ( g ) k B ( g ) − B ( g ) B ( k ) B ( g ) − = e , that is, B ( g ) k B ( g ) − is in K − . Thus K − ⊂ G − is a normal Lie subgroup.Likewise we can prove that K + is a normal Lie subgroup of G + . (cid:3) Based on Lemma 3.2, we define a map Θ : G − / K − → G + / K + , Θ ( B ( g )) = B + ( g ) , ∀ g ∈ G , where · denotes the equivalence class in the two quotients. To see that Θ is well-defined, let k ∈ K − . We have k = B ( k ) − = B ( k † ). Note that k † = Ad B ( k ) − k − = k − . We have Θ ( B ( g ) k ) = Θ ( B ( g ∗ k − )) = B + ( g ∗ k − ) = g ∗ k − B ( g ∗ k − ) = g B ( g ) k − B ( g ) − B ( g ) B ( k − ) = g B ( g ) = Θ ( B ( g )) , which implies that Θ is well-defined. Proposition 3.3.
The map Θ : G − / K − → G + / K + is a Lie group isomorphism, and is called the Cayley transform of the Rota-Baxter operator B . The term Cayley transform is adapted from a similar notion from r -matrices [15, 38]. NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 13
Proof.
It is obvious that Θ is surjective. To see that it is also injective, if B + ( g ) = g B ( g ) ∈ K + ,that is, B ( g B ( g )) = e , then we have B + ( B ( g )) = B ( g ) B ( B ( g )) = B ( g ∗ B ( g )) = B ( g Ad B ( g ) B ( g )) = B ( g B ( g )) = e , which implies that B ( g ) ∈ K − . This proves that Θ is injective.We next show that Θ is a Lie group homomorphism, which follows from Θ ( B ( g ) B ( g ′ )) = Θ ( B ( g ∗ g ′ )) = B + ( g ∗ g ′ ) = B + ( g ) B + ( g ′ ) = Θ ( B ( g )) Θ ( B ( g )) , by Proposition 3.1. Therefore, Θ is a Lie group isomorphism. (cid:3) Now we consider the product Lie group ( G + × G − , · D ), whose Lie group product is( g + , g − ) · D ( h + , h − ) : = ( g + h + , g − h − ) , ∀ g + , h + ∈ G + , g − , h − ∈ G − . Let G Θ ⊂ G + × G − denote the subset G Θ : = { ( g + , g − ) ∈ G + × G − ; Θ ( g − ) = g + } . Define a map Φ : G → G Θ by Φ ( g ) : = ( B + ( g ) , B ( g )) . Lemma 3.4.
With the above notations, G Θ is a Lie subgroup of ( G + × G − , · D ) . Moreover, the map Φ is a Lie group isomorphism from ( G , ∗ ) to G Θ .Proof. By Proposition 3.3, for any ( g + , g − ) , ( h + , h − ) ∈ G Θ , we have Θ ( g − h − ) = Θ ( g − h − ) = Θ ( g − ) Θ ( h − ) = g + h + = g + h + , which implies that ( g + h + , g − h − ) ∈ G Θ and then G Θ is a subgroup of ( G + × G − , · D ).We next check that Φ is a bijection. Let g ∈ G such that Φ ( g ) = ( e , e ). Then we have B ( g ) = g B ( g ) = e . Thus g = e , which implies that Φ is injective. For any ( g + , g − ) ∈ G Θ , we have Θ ( g − ) = g + . Since g − ∈ G − , there exists g ∈ G such that B ( g ) = g − . Hence we get Θ ( g − ) = Θ ( B ( g )) = g B ( g ) . Therefore, g + = g B ( g ), which means that there exists k ∈ K + such that g + = g B ( g ) k . Let g ′ = g ∗ k . Then we have Φ ( g ′ ) = ( g ∗ k B ( g ∗ k ) , B ( g ∗ k )) = ( g B ( g ) k B ( g ) − B ( g ) , B ( g )) = ( g + , g − ) . Therefore, Φ is surjective.Finally, for any g , h ∈ G , by Proposition 3.1, we have Φ ( g ∗ h ) = ( B + ( g ∗ h ) , B ( g ∗ h )) = ( B + ( g ) B + ( h ) , B ( g ) B ( h )) = ( B + ( g ) , B ( g )) · D ( B + ( h ) , B ( h )) = Φ ( g ) · D Φ ( h ) . Therefore, Φ is a Lie group homomorphism and hence a Lie group isomorphism. (cid:3) Theorem 3.5. (Factorization theorem of Rota-Baxter Lie groups) Let ( G , B ) be a Rota-BaxterLie group. Then every element g ∈ G can be uniquely expressed as g = g + g − − for ( g + , g − ) ∈ G Θ . Proof.
For any g ∈ G , we have g = B + ( g ) B ( g ) − . To see the uniqueness, if g = g + g − − = h + h − − ,then h − + g + = h − − g − ∈ G + ∩ G − and Θ ( h − − g − ) = h − + g + . Suppose h − + g + = h − − g − = B ( s ) ∈ G + ∩ G − for some s ∈ G . Then Θ ( B ( s )) = B + ( s ) = s B ( s ) = B ( s ) , which implies that s B ( s ) = B ( s ) k for some k ∈ K + . Note that K + ⊂ G + is a normal subgroup. Wehave s = B ( s ) k B ( s ) − ∈ K + . Therefore, B ( s ) = e . Hence we get that h + = g + and h − = g − . (cid:3) Conversely, we obtain the following generalization of Lemma 2.7 by the same argument.
Proposition 3.6.
Suppose that G is a Lie group together with two Lie subgroups G ± . Let ˜ G ⊂ G + × G − be a Lie subgroup. Assume that each element g ∈ G can be uniquely decomposed asg = g + g − − , ( g + , g − ) ∈ ˜ G . (17) Then B : G → G defined by B ( g + g − − ) = g − is a Rota-Baxter operator on the Lie group G. At the end of this section, we give some remarks on the applications of the factorization theo-rem of Rota-Baxter Lie groups to the factorization theorems of Semenov-Tian-Shansky [38, 39].We refer the reader to the original references for the related background.
Remark 3.7. (i) Applying di ff erentiation, Theorem 3.5 gives a factorization of Rota-BaxterLie algebras of weight 1, whose form is similar to the Atkinson factorization of Rota-Baxter associative algebras [1, 17]. As noted in the introduction, B is a Rota-Baxteroperator of weight 1 on a Lie algebra ( g , [ · , · ] g ) if and only if R : = id + B satisfies themodified Yang-Baxter equation in (2). Then it is straightforward to see that the abovefactorization of Rota-Baxter Lie algebras of weight 1 coincides with the InfinitesimalFactorization Theorem of Semenov-Tian-Shansky for Lie algebras [38, Prop. 9].(ii) Integrating the above Infinitesimal Factorization Theorem of Lie algebras, Semenov-Tian-Shansky obtained his famous Global Factorization Theorem for Lie groups, heldin a small enough neighborhood of the unit. This is a fundamental tool in studyingintegrable systems. See [38, Theorem. 11] and [39, Theorem. 3.3] for more details. OurTheorem 3.5 shows that the Global Factorization Theorem can be derived directly on thegroup level and is a truly global result since it applies to all elements of the Lie group.In summary, we have the following commutative diagram on the relationship between the in-finitesimal and global versions of the Factorization Theorem, where the classical approach tookthe right-down path and our approach takes the down-right path.( g , R ) factorization / / R = id + B (cid:15) (cid:15) g θ ⊆ g + ⊕ g − ( g , B ) factorization / / integration (cid:15) (cid:15) g θ ⊆ g + ⊕ g − integration (cid:15) (cid:15) ( G , B ) factorization / / G Θ ⊆ G + × G − . (18)Here the factorizations in the first two rows are the Infinitesimal Factorization Theorem for ( g , R )from the modified Yang-Baxter equation by Semenov-Tian-Shansky and its equivalent form forthe Rota-Baxter Lie algebra ( g , B ) in Remark 3.7(i). NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 15
4. R ota -B axter L ie algebroids and R ota -B axter L ie groupoids In this section, we introduce the notions of Rota-Baxter Lie algebroids and Rota-Baxter Liegroupoids. We show that a Rota-Baxter Lie algebroid gives rise to a post-Lie algebroid and that aRota-Baxter Lie algebroid can be obtained from a Rota-Baxter groupoid by di ff erentiation. More-over, actions of Rota-Baxter Lie algebras, actions of Rota-Baxter Lie groups and actions of post-Lie algebras are introduced to produce Rota-Baxter Lie algebroids, Rota-Baxter Lie groupoidsand post-Lie algebroids.4.1. Rota-Baxter Lie algebroids.
We introduce the notion of a Rota-Baxter Lie algebroid andshow that an action of a Rota-Baxter Lie algebra naturally gives rise to a Rota-Baxter Lie alge-broid.
Definition 4.1. ([27]) A
Lie algebroid structure on a vector bundle
A −→ M is a pair consistingof a Lie algebra structure [ · , · ] A on the section space Γ ( A ) and a vector bundle morphism a A : A −→
T M from A to the tangent bundle T M , called the anchor , satisfying the relation(19) [ x , f y ] A = f [ x , y ] A + a A ( x )( f ) y , ∀ x , y ∈ Γ ( A ) , f ∈ C ∞ ( M ) . When a A is surjective, we call A a transitive Lie algebroid .We usually denote a Lie algebroid by ( A , [ · , · ] A , a A ) or simply A if there is no danger ofconfusion.We now introduce the notion of a Rota-Baxter operator on a Lie algebroid. Definition 4.2.
Given a scalar λ , a Rota-Baxter operator of weight λ on a transitive Lie alge-broid ( A , [ · , · ] A , a A ) is a bundle map B : ker( a A ) −→ A covering the identity such that[ B ( u ) , B ( v )] A = B ([ B ( u ) , v ] A ) + B ([ u , B ( v )] A ) + λ B ([ u , v ] A ) , ∀ u , v ∈ Γ (ker( a A )) . (20)A Rota-Baxter Lie algebroid is a Lie algebroid equipped with a Rota-Baxter operator of weight1. Since ker( a A ) is an ideal of A , the terms in (20) are well-defined. Remark 4.3.
Here we use ker( a A ) instead of A to ensure that Eq. (20) is compatible with thefunction linearity of B , that is, B ( f u ) = f B ( u ) for any function f ∈ C ∞ ( M ). Indeed, imposingthe function linearity for a bundle map B : A → A in the definition of a Rota-Baxter operatorof weight λ on a Lie algebroid ( A , [ · , · ] A , a A ), we obtain0 = [ B ( f u ) , B ( v )] A − B ([ B ( f u ) , v ] A + [ f u , B ( v )] A + λ [ f u , v ] A ) = [ f B ( u ) , B ( v )] A − B ([ f B ( u ) , v ] A + [ f u , B ( v )] A + λ [ f u , v ] A ) = f [ B ( u ) , B ( v )] A − a A ( B ( v ))( f ) B ( u ) − f B [ B ( u ) , v ] A + a A ( v )( f ) B ( u ) − f B [ u , B ( v )] A + a A ( B ( v ))( f ) B ( u ) − λ f B [ u , v ] A + λ a A ( v )( f ) B ( u ) = a A ( v )( f ) B ( u ) + λ a A ( v )( f ) B ( u ) . Therefore, B needs to satisfy the additional condition a A ( v )( f ) B ( u ) + λ a A ( v )( f ) B ( u ) = , ∀ u , v ∈ Γ ( A ) , f ∈ C ∞ ( M ) . Thus, it is natural to restrict the domain of B to ker( a A ). Remark 4.4.
A vector space is a vector bundle over a point. Therefore, a Lie algebra is naturallya Lie algebroid with the anchor being zero. It is obvious that a Rota-Baxter Lie algebroid reducesto a Rota-Baxter Lie algebra when the underlying Lie algebroid reduces to a Lie algebra.We first give some simple examples of Rota-Baxter Lie algebroids.
Example 4.5.
Let ( g , [ · , · ] g , B ) be a Rota-Baxter Lie algebra of weight 1, and M a manifold.Consider the trivial bundle M × g . Then the linear map B : g → g naturally gives rise to abundle map B : M × g → M × g , ( m , u ) ( m , B ( u )). Furthermore, the Lie bracket [ · , · ] g canbe naturally extended to Γ ( M × g ) = C ∞ ( M ) ⊗ g , which is function linear. Then it is obvious that( M × g , [ · , · ] g , B ) is a Rota-Baxter Lie algebroid with the anchor being zero. Example 4.6.
Let ( A , [ · , · ] A , a A ) be a transitive Lie algebroid. The minus of the inclusion B : ker( a A )
7→ A , B ( u ) = − u , is naturally a Rota-Baxter operator of weight 1 on A .Now we construct a class of examples of particular interest, namely the action Rota-Baxter Liealgebroids.First we recall actions of Lie algebras on manifolds and the associated action Lie algebroids([27]). Let φ : g −→ X ( M ) be a left action of a Lie algebra g on a manifold M , that is, a Lie algebrahomomorphism from ( g , [ · , · ] g ) to the Lie algebra ( X ( M ) , [ · , · ] T M ) of vector fields. Then we have aLie algebroid structure on the trivial bundle A = M × g , whose anchor a A : M × g → T M andLie bracket [ · , · ] A : ∧ ( C ∞ ( M ) ⊗ g ) → C ∞ ( M ) ⊗ g are given by a A ( m , u ) = φ ( u ) m , ∀ m ∈ M , u ∈ g , (21) [ f u , gv ] A = f g [ u , v ] g + f φ ( u )( g ) v − g φ ( v )( f ) u , ∀ u , v ∈ g , f , g ∈ C ∞ ( M ) . (22)This Lie algebroid is called the action Lie algebroid of the Lie algebra ( g , [ · , · ] g ) and the action φ , and denoted by g × φ M . If φ is a transitive action, then g × φ M is a transitive Lie algebroid. Proposition 4.7.
With the above notations, a bundle map B : ker( a A ) → g × φ M gives a Rota-Baxter operator of weight on g × φ M if and only if B m : = B| m : ker( a A ) m → g for any m ∈ Msatisfies that [ B m ( u ) , B m ( v )] g = B m ([ B m ( u ) , v ] g + [ u , B m ( v )] g + [ u , v ] g ) , ∀ u , v ∈ ker( a A ) m . Proof.
It follows from a straightforward verification. (cid:3)
Definition 4.8. An action of a Rota-Baxter Lie algebra ( g , [ · , · ] g , B ) on a manifold M is a ho-momorphism of Lie algebras φ : ( g , [ · , · ] B ) → X ( M ), where the Lie bracket [ · , · ] B is defined by(13).Let φ : ( g , [ · , · ] B ) → X ( M ) be an action of the Rota-Baxter Lie algebra ( g , [ · , · ] g , B ) on amanifold M . Consider the direct sum bundle A : = ( M × g ) ⊕ T M . Then Γ ( A ) = ( C ∞ ( M ) ⊗ g ) ⊕ X ( M ). There is naturally a Lie algebroid structure on A whose anchor a A is the projection to T M and whose bracket is determined by[ f u + X , gv + Y ] A : = f g [ u , v ] g + X ( g ) v − Y ( f ) u + [ X , Y ] T M , for all X , Y ∈ X ( M ) , u , v ∈ g , f , g ∈ C ∞ ( M ).Consider the bundle map B : ker( a A ) = M × g −→ ( M × g ) ⊕ T M defined by(23) B ( m , u ) : = ( m , B ( u ) , φ ( u )( m )) , ∀ m ∈ M , u ∈ g . NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 17
Proposition 4.9.
With the above notations, the bundle map B defined by (23) is a Rota-Baxteroperator on the Lie algebroid (( M × g ) ⊕ T M , [ · , · ] A , a A ) . The Rota-Baxter algebroid (( M × g ) ⊕ T M , [ · , · ] A , a A , B ) will be called the action Rota-Baxteralgebroid in the sequel. Proof.
The bundle map B is a Rota-Baxter operator of weight 1 on A , that is, (20) holds for λ =
1, if and only if[ B ( u ) , B ( v )] g = B ([ B ( u ) , v ] g + [ u , B ( v )] g + [ u , v ] g ) , ∀ u , v ∈ g , [ φ ( u ) , φ ( v )] T M = φ ([ B ( u ) , v ] g + [ u , B ( v )] g + [ u , v ] g ) . That is, B is a Rota-Baxter operator of weight 1 on the Lie algebra ( g , [ · , · ] g ) and φ is a left actionof the Lie algebra ( g , [ · , · ] B ) on M . (cid:3) Relations to post-Lie algebroids.
A notion that is closely related to a Rota-Baxter Lie al-gebra is the post-Lie algebra. We recall some background in order to present a more generalrelationship between Rota-Baxter Lie algebroids and post-Lie algebroids. Moreover, we intro-duce the notion of actions of post-Lie algebras on manifolds, which produce a class of interestingpost-Lie algebroids.
Definition 4.10. ([41]) A post-Lie algebra ( g , [ · , · ] g , ⊲ ) consists of a Lie algebra ( g , [ · , · ] g ) and abinary product ⊲ : g ⊗ g → g such that u ⊲ [ v , w ] g = [ u ⊲ v , w ] g + [ v , u ⊲ w ] g , (24) [ u , v ] g ⊲ w = a ⊲ ( u , v , w ) − a ⊲ ( v , u , w ) , (25)here a ⊲ ( u , v , w ) : = u ⊲ ( v ⊲ w ) − ( u ⊲ v ) ⊲ w and u , v , w ∈ g . Define L ⊲ : g → gl ( g ) by L ⊲ ( u )( v ) = u ⊲ v . Then by (24), L ⊲ is a linear map from g to Der( g ). Remark 4.11.
Let ( g , [ · , · ] g , ⊲ ) be a post-Lie algebra. If the Lie bracket [ · , · ] g =
0, then ( g , ⊲ )becomes a pre-Lie algebra. Thus, a post-Lie algebra can be viewed as a nonabelian version of apre-Lie algebra. See [8, 9, 10] for the classifications of post-Lie algebras on certain Lie algebras,and [3, 28] for applications of post-Lie algebras in integrable systems and numerical integrations.The notion of a post-Lie algebroid was given in [28] in the study of geometric numerical anal-ysis. Post-Lie algebroids are geometrization of post-Lie algebras. See [29] for more applicationsof post-Lie algebroids. Definition 4.12. A post-Lie algebroid structure on a vector bundle A −→ M is a triple thatconsists of a C ∞ ( M )-linear Lie algebra structure [ · , · ] A on Γ ( A ), a bilinear operation ⊲ A : Γ ( A ) × Γ ( A ) −→ Γ ( A ) and a vector bundle morphism a A : A −→ T M , called the anchor , such that( Γ ( A ) , [ · , · ] A , ⊲ A ) is a post-Lie algebra, and for all f ∈ C ∞ ( M ) and u , v ∈ Γ ( A ), the followingrelations are satisfied:(i) u ⊲ A ( f v ) = f ( u ⊲ A v ) + a A ( u )( f ) v , (ii) ( f u ) ⊲ A v = f ( u ⊲ A v ) . We usually denote a post-Lie algebroid by ( A , [ · , · ] A , ⊲ A , a A ). If the Lie algebra structure [ · , · ] A in a post-Lie algebroid ( A , [ · , · ] A , ⊲ A , a A ) is abelian, then it becomes a left-symmetric algebroid,which is also called a Koszul-Vinberg algebroid. See [25, 26, 30, 31] for more details. A Rota-Baxter Lie algebra gives rise to a post-Lie algebra [2]. More precisely, let ( g , [ · , · ] g , B )be a Rota-Baxter Lie algebra. Then for ⊲ defined by u ⊲ v = [ B ( u ) , v ] g , ( g , [ · , · ] g , ⊲ ) is a post-Lie algebra. We call it the splitting post-Lie algebra of the Rota-BaxterLie algebra ( g , B ) since this is a special case of the process of the splitting of an operad with aRota-Baxter operator [2]. As the geometrization of this fact, we have Theorem 4.13.
Let ( A , [ · , · ] A , a A , B ) be a Rota-Baxter Lie algebroid. Let A = ker( a A ) . Then for ⊲ A defined by u ⊲ A v : = [ B ( u ) , v ] A , ∀ u , v ∈ Γ (ker( a A )) , (26)( A , [ · , · ] A , ⊲ A , a A ◦ B ) is a post-Lie algebroid, called the splitting post-Lie algebroid of the Rota-Baxter Lie algebroid ( A , [ · , · ] A , a A , B ) .Proof. By (19), the operation [ · , · ] A on Γ (ker( a A )) is C ∞ ( M )-linear. By (20), we can deduce that(ker( a A ) , [ · , · ] A , ⊲ A ) is a post-Lie algebra. Finally, for all u , v ∈ Γ (ker( a A )) and f ∈ C ∞ ( M ), wehave ( f u ) ⊲ A v = [ f B ( u ) , v ] A = f [ B ( u ) , v ] A = f u ⊲ A v , u ⊲ A ( f v ) = [ B ( u ) , f v ] A = f [ B ( u ) , v ] A + a A ◦ B ( u )( f ) v = f u ⊲ A v + a A ◦ B ( u )( f ) v , which implies that (ker( a A ) , [ · , · ] A , ⊲ A , a A ◦ B ) is a post-Lie algebroid. (cid:3) Example 4.14.
Let ( g , [ · , · ] g , ⊲ ) be a post-Lie algebra. Consider the trivial bundle A = M × g . Then the operation ⊲ can be naturally extended to ⊲ A defined on Γ ( M × g ) = C ∞ ( M ) ⊗ g , which isfunction linear. The Lie bracket [ · , · ] g can be similarly extended. It is obvious that ( M × g , [ · , · ] g , ⊲ A )is naturally a post-Lie algebroid with the anchor map being zero. We call it the post-Lie algebrabundle . Example 4.15.
The induced post-Lie algebroid of the Rota-Baxter operator given in Example 4.6is (ker( a A ) , [ · , · ] A , ⊲ A , u ⊲ A v : = − [ u , v ] A , ∀ u , v ∈ Γ (ker( a A )) . In the sequel, we introduce actions of post-Lie algebras on manifolds, which will produce aclass of interesting examples of post-Lie algebroids. Recall that a post-Lie algebra ( g , [ · , · ] g , ⊲ )gives rise to a new subjacent Lie bracket ~ · , · (cid:127) :(27) ~ u , v (cid:127) = [ u , v ] g + u ⊲ v − v ⊲ u . Definition 4.16. An action of a post-Lie algebra ( g , [ · , · ] g , ⊲ ) on a manifold M is homomorphismof Lie algebras φ : ( g , ~ · , · (cid:127) ) → X ( M ) . Remark 4.17.
If the post-Lie algebra reduces to a pre-Lie algebra, then the above definitionreduces to the definition of actions of pre-Lie algebras on manifolds, which was given in [26] forthe study of left-symmetric algebroids.It is straightforward to see that an action of a Rota-Baxter Lie algebra also gives an action ofthe splitting post-Lie algebra.
Lemma 4.18.
Let φ : ( g , [ · , · ] B ) → X ( M ) be an action of the Rota-Baxter Lie algebra ( g , [ · , · ] g , B ) on a manifold M. Then φ is also an action of the splitting post-Lie algebra ( g , [ · , · ] g , ⊲ ) from theRota-Baxter Lie algebra. NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 19
Let φ : ( g , ~ · , · (cid:127) ) → X ( M ) be an action of the post-Lie algebra ( g , [ · , · ] g , ⊲ ) on a manifold M .On the trivial bundle A = M × g , define the anchor a A : M × g → T M and the bilinear operation ⊲ A : ⊗ ( C ∞ ( M ) ⊗ g ) → C ∞ ( M ) ⊗ g by a A ( m , u ) : = φ ( u ) m , ∀ m ∈ M , u ∈ g , (28) ( f u ) ⊲ A gv : = f gu ⊲ v + f φ ( u )( g ) v , ∀ u , v ∈ g , f , g ∈ C ∞ ( M ) . (29) Proposition 4.19.
With the above notations, ( M × g , [ · , · ] g , ⊲ A , a A ) is a post-Lie algebroid, calledthe action post-Lie algebroid of the post-Lie algebra ( g , [ · , · ] g , ⊲ ) .Proof. It follows from a direct verification. (cid:3)
Consider the action Rota-Baxter Lie algebroid given in Proposition 4.9. By Theorem 4.13,there is a splitting post-Lie algebroid. It turns out that this post-Lie algebroid is the action post-Lie algebroid.
Corollary 4.20.
Let φ : ( g , [ · , · ] B ) → X ( M ) be an action of a Rota-Baxter Lie algebra ( g , [ · , · ] g , B ) on a manifold M. Let (( M × g ) ⊕ T M , [ · , · ] A , a A , B ) be the action Rota-Baxter Lie algebroid givenin Proposition 4.9. Then the splitting post-Lie algebroid of (( M × g ) ⊕ T M , [ · , · ] A , a A , B ) is exactlythe action post-Lie algebroid of the splitting post-Lie algebra ( g , [ · , · ] g , ⊲ ) of the Rota-Baxter Liealgebra ( g , [ · , · ] g , B ) .Proof. By Lemma 4.18, φ : ( g , [ · , · ] B ) → X ( M ) is also an action of the underlying post-Liealgebra ( g , [ · , · ] g , ⊲ ) on M . By Theorem 4.13, the induced post-Lie algebroid is given by ( M × g , [ · , · ] g , ⊲ A , a A ), where for all f , g ∈ C ∞ ( M ) , u , v ∈ g ,( f u ) ⊲ A ( gv ) : = [ f B ( u ) + f φ ( u ) , gv ] A = f g [ B ( u ) , v ] g + f φ ( u )( g ) v = f gu ⊲ v + f φ ( u )( g ) v , a A ( u , m ) : = a A ◦ B ( u , m ) = φ ( u )( m ) , which is exactly the action post-Lie algebroid of the post-Lie algebra ( g , [ · , · ] g , ⊲ ) underlying theRota-Baxter Lie algebra ( g , [ · , · ] g , B ). (cid:3) Remark 4.21.
A post-Lie algebra ( g , [ · , · ] g , ⊲ ) gives rise to a subjacent Lie algebra ( g , ~ · , · (cid:127) ) inEq (27). Similarly, it was shown in [28] that a post-Lie algebroid ( A , [ · , · ] A , ⊲ A , a A ) gives rise to a subjacent Lie algebroid ( A , ~ · , · (cid:127) A , a A ), where the Lie bracket ~ · , · (cid:127) A is given by ~ u , v (cid:127) A = [ u , v ] A + u ⊲ A v − v ⊲ A u . It is straightforward to see that when the post-Lie algebroid comes from the action φ of a post-Liealgebra ( g , [ · , · ] g , ⊲ ), this subjacent Lie algebroid ( A , ~ · , · (cid:127) A , a A ) is exactly the action Lie algebroidof the subjacent Lie algebra ( g , ~ · , · (cid:127) ). See [29, Theorem 4.4] for the necessary and su ffi cientconditions for a Lie algebroid admitting a post-Lie algebroid structure.By Corollary 4.20 and Remark 4.21, we have the following commutative diagram:Rota-Baxter Lie algebra splitting / / action (cid:15) (cid:15) post-Lie algebra action (cid:15) (cid:15) subjacent / / Lie algebra action (cid:15) (cid:15)
Rota-Baxter Lie algebroid splitting / / post-Lie algebroid subjacent / / Lie algebroid.(30)
Rota-Baxter Lie groupoids and integration of Rota-Baxter Lie algebroids.
We now in-troduce the notion of a Rota-Baxter Lie groupoid and show that the di ff erentiation of a Rota-Baxter Lie groupoid is a Rota-Baxter Lie algebroid. Moreover, we introduce the notion of actionsof Rota-Baxter Lie groups, from which we construct action Rota-Baxter Lie groupoids.Recall that a groupoid [27] is a small category such that every arrow is invertible. Explicitly, Definition 4.22. A groupoid is a pair ( G , M ), where M is the set of objects and G is the set ofarrows, with the structure maps • two surjective maps s , t : G −→ M , called the source map and target map, respectively; • the multiplication · : G (2) −→ G , where G (2) = { ( g , g ) ∈ G × G | s ( g ) = t ( g ) } ; • the inverse map ( · ) − : G −→ G ; • the inclusion map ι : M −→ G , called the identity map;satisfying the following properties:(i) (associativity) ( g · g ) · g = g · ( g · g ), whenever the multiplications are well-defined;(ii) (unitality) ι ( t ( g )) · g = g = g · ι ( s ( g ));(iii) (invertibility) g · g − = ι ( t ( g )), g − · g = ι ( s ( g )).We also denote a groupoid by ( G ⇒ M , s , t ) or simply by G .A Lie groupoid is a groupoid such that both the set of objects and the set of arrows are smoothmanifolds, all structure maps are smooth, and the source and target maps are surjective submer-sions.The tangent space of a Lie group at the identity has a Lie algebra structure. As its geometriza-tion, on the vector bundle A : = ker( t ∗ ) | M −→ M from a Lie groupoid, there is a Lie algebroidstructure defined as follows ([27]): the anchor map a A : A −→
T M is simply s ∗ and the Liebracket [ u , v ] A is determined by ←−−−−− [ u , v ] A = − [ ←− u , ←− v ] T G , ∀ u , v ∈ Γ ( A ) , where ←− u denotes the left-invariant vector field on G given by ←− u g = L g ∗ u s ( g ) .Denote by O m : = s ◦ t − ( m ) for m ∈ M . When O m = M for m ∈ M , we call the Lie groupoid a transitive Lie groupoid . The Lie algebroid associated to a transitive Lie groupoid is a transitiveLie algebroid.Let G ⇒ M be a transitive Lie groupoid. Its isotropy group at m ∈ M is defined to be H m : = s − ( m ) ∩ t − ( m ) . Denote by H the bundle of Lie groups over M whose fiber at m is the Lie group H m . Definition 4.23. A Rota-Baxter operator on a transitive Lie groupoid G ⇒ M is a map B : H → G covering the identity map on M with respect to the target map, that is, t ◦ B = t ,satisfying(31) B ( g ) B ( h ) = B ( g Ad B ( g ) h ) , ∀ g , h ∈ H , such that s ( B ( g )) = t ( h ) . The requirements t ( B ( g )) = t ( g ) , t ( B ( h )) = t ( h ) and s ( B ( g )) = t ( h ) are to ensure that themultiplications appeared in the above formula are well defined. NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 21
Proposition 4.24.
A Rota-Baxter operator B on a Lie groupoid G associates with H a new Liegroupoid structure, called the descendent Lie groupoid , given by e s ( g ) : = s ( B ( g )) , e t ( g ) : = t ( g ) , g ⋆ h : = g Ad B ( g ) h , ∀ g , h ∈ H s . t . s ( B ( g )) = t ( h ) . Proof.
It follows from a direct verification. (cid:3)
Denote this Lie groupoid by ( H , ⋆ ). Theorem 4.25.
Let ( G , B ) be a transitive Rota-Baxter Lie groupoid. Let A = ker( t ∗ ) | M be the Liealgebroid of G and B : = B ∗ : ker( a A ) −→ A the tangent map of B at the identity. Then ( A , B ) isa Rota-Baxter Lie algebroid.Proof. We first claim that the Lie algebroid of ( H , ⋆ ) is (ker( a A ) , [ · , · ] B , a A ◦ B ), where[ u , v ] B = [ B ( u ) , v ] A + [ u , B ( v )] A + [ u , v ] A , ∀ u , v ∈ Γ (ker( a A )) . Note that ker(˜ t ∗ ) | M = ker( t ∗ ) | M = ker( a A ) and ˜ s ∗ = s ∗ ◦ B . Then following the same proof as inProposition 2.14, we find that the Lie bracket on ker( a A ) of the Lie groupoid structure ⋆ is [ · , · ] B .Moreover, it is direct to show that B : ( H , ⋆ ) → G is a Lie groupoid homomorphism, whichinduces a Lie algebroid homomorphism B : (ker( a A ) , [ · , · ] B , a A ◦ B ) → A . This implies that[ B ( u ) , B ( v )] A = B ([ B ( u ) , v ] A + [ u , B ( v )] A + [ u , v ] A ) . So ( A , B ) is a Rota-Baxter Lie algebroid. (cid:3) Assume that a Lie group G left acts on a manifold M . Then we obtain a Lie groupoid G × M ⇒ M , whose source, target maps and multiplication are: s ( g , m ) : = m , t ( g , m ) : = g · m , ( h , n )( g , m ) : = ( hg , m ) , for n = g · m . This Lie groupoid is called the action Lie groupoid , whose Lie algebroid isthe action Lie algebroid. Make the above phrase more precise, so that we have a commutativediagram actions of Lie groups on M action / / di ff erentiation (cid:15) (cid:15) Lie groupoids di ff erentiation (cid:15) (cid:15) actions of Lie algebras on M action / / Lie algebroids.(32)Suppose that the action is transitive. The isotropy group at m ∈ M is H m = { g ∈ G | g · m = m } . Consider the Rota-Baxter operator on the action Lie groupoid.
Proposition 4.26.
A map B : H → G × M is a Rota-Baxter operator of weight on the actionLie groupoid if and only if B m : H m → G × { m } (cid:27) G given by B ( g , m ) = ( B m ( g ) , B m ( g ) − m ) satisfies B m ( g ) B n ( h ) = B m ( g Ad B m ( g ) h ) , ∀ g ∈ H m , h ∈ H n , n = B m ( g ) − m . (33)Taking the di ff erential, we get Proposition 4.7. Proof.
In fact, let ˜ g = ( g , m ) ∈ H m and ˜ h = ( h , n ) ∈ H n such that s ( B ( ˜ g )) = B m ( g ) − m = t (˜ h ) = n . Note that gm = m and hn = n . Then we have B ( ˜ g ) B (˜ h ) = ( B m ( g ) , B m ( g ) − m )( B n ( h ) , B n ( h ) − n ) = ( B m ( g ) B n ( h ) , B n ( h ) − n )and B ( ˜ g Ad B (˜ g ) ˜ h ) = B (( g , m )( B m ( g ) , B m ( g ) − m )( h , n )( B m ( g ) − , m )) = B (( g B m ( g ) h B m ( g ) − , m )) = ( B m ( g Ad B m ( g ) h ) , B m ( g Ad B m ( g ) h ) − m ) . Thus, B is a Rota-Baxter operator if and only if (33) holds. (cid:3) Finally we study the groupoid analog of the action Rota-Baxter Lie algebroid given in Propo-sition 4.9. We give the definition of actions of Rota-Baxter Lie groups as follows.
Definition 4.27. An action of a Rota-Baxter Lie group ( G , B ) on a manifold M is defined tobe a right action B of the descendent Lie group ( G , ∗ ) on M , where the group multiplication ∗ isgiven by (14).Let B be an action of a Rota-Baxter Lie group ( G , B ) on a manifold M . Consider the Liegroupoid G : = M × G × M ⇒ M , where the source, target and the multiplication of this Liegroupoid are given by s ( m , g , n ) : = n , t ( m , g , n ) : = m , ( m ′ , h , n ′ ) · ( m , g , n ) : = ( m ′ , hg , n ) , when n ′ = m . It is easy to see that the bundle of isotropy groups is G × M → M . The inclusion of G × M in G is given by ( g , m ) ֒ → ( m , g , m ).Define a map B : G × M → M × G × M by B ( g , m ) = ( m , B ( g ) , B ( g , m )) . Proposition 4.28.
Let B be an action of a Rota-Baxter Lie group ( G , B ) on a manifold M. Thenthe map B defined above is a Rota-Baxter operator on the Lie groupoid G = M × G × M ⇒ M. This Rota-Baxter Lie groupoid is called the action Rota-Baxter Lie groupoid . Proof.
Let ˜ g = ( g , m ) , ˜ h = ( h , n ) ∈ G × M such that s ( B ( ˜ g )) = B ( g , m ) = t (˜ h ) = n . Then we have B ( ˜ g ) B (˜ h ) = ( m , B ( g ) , B ( g , m ))( n , B ( h ) , B ( h , n )) = ( m , B ( g ) B ( h ) , B ( h , B ( g , m )))and B ( ˜ g Ad B (˜ g ) ˜ h ) = B (( m , g , m )( m , B ( g ) , B ( g , m ))( n , h , n )( B ( g , m ) , B ( g ) − , m )) = B (( m , g B ( g ) h B ( g ) − , m )) = ( m , B ( g Ad B ( g ) h ) , B ( g Ad B ( g ) h , m )) . Thus, B ( ˜ g ) B (˜ h ) = B ( ˜ g Ad B (˜ g ) ˜ h ) holds if and only if B ( g ) B ( h ) = B ( g Ad B ( g ) h ) , B ( h , B ( g , m )) = B ( g Ad B ( g ) h , m ) , NTEGRATION AND GEOMETRIZATION OF ROTA-BAXTER LIE ALGEBRAS 23 which implies that B is a Rota-Baxter operator on the Lie group G and B defines a right actionof the Lie group ( G , ∗ ) on M . (cid:3) In summary, the diagram in (32) can be enriched to the diagramactions of Rota-Baxter Lie groups on M action / / di ff erentiation (cid:15) (cid:15) Rota-Baxter Lie groupoids di ff erentiation (cid:15) (cid:15) actions of Rota-Baxter Lie algebras on M action / / Rota-Baxter Lie algebroids.
Acknowledgements.
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