We study multiplicative Dirac structures on Lie groups. We show that the
characteristic foliation of a multiplicative Dirac structure is given by the
cosets of a normal Lie subgroup and, whenever this subgroup is closed, the leaf
space inherits the structure of a Poisson-Lie group. We also describe
multiplicative Dirac structures on Lie groups infinitesimally.Comment: Published in Comptes Rendus Mathematiqu

Bursztyn

Courant

Cristián Ortiz

Drinfeld

Mackenzie

Semenov-Tian-Shansky

English

arXiv

Ortiz, Cristián

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1279–1282http://france.elsevier.com/direct/CRASS1/Differential GeometryMultiplicative Dirac structures on Lie groupsCristián Ortiz 1Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro 22460-320, BrazilReceived 18 February 2008; accepted after revision 8 October 2008Available online 1 November 2008Presented by Charles-Michel MarleAbstractWe study multiplicative Dirac structures on Lie groups. We show that the characteristic foliation of a multiplicative Diracstructure is given by the cosets of a normal Lie subgroup and, whenever this subgroup is closed, the leaf space inherits the structureof a Poisson–Lie group. We also describe multiplicative Dirac structures on Lie groups infinitesimally. To cite this article: C. Ortiz,C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméStructures de Dirac multiplicatives sur les groupes de Lie. Nous étudions les structures de Dirac multiplicatives sur lesgroupes de Lie. On montre que le feuilletage caractéristique d’une structure de Dirac multiplicative est donnée par les classes àgauche (respectivement à droite) d’un sous-groupe distingué et, quand ce sous-groupe est fermé, l’espace des feuilles est munid’une structure de groupe de Lie–Poisson. Nous décrivons aussi la version infinitésimale des structures de Dirac multiplicatives surles groupes de Lie. Pour citer cet article : C. Ortiz, C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeUn groupe de Lie–Poisson est un objet du type groupe de Lie dans la catégorie des variétés de Poisson, c’est-à-direun groupe de Lie G muni d’une structure de Poisson π ∈ Γ (∧2T G) telle que la multiplication m :G × G → G soitune application de Poisson, le produit G × G étant muni de la structure de Poisson produit. De manière équivalente,le tenseur de Poisson satisfait la condition multiplicative πgh = (Lg)∗πh + (Rh)∗πg . Les groupes de Lie–Poissonont été introduits par Drinfeld [4]. Ces structures apparaissent aussi dans l’etude des propriétés Hamiltonnienes destransformations d’habillage de certain systèmes intégrables [7].La donnée infinitésimale associée à un groupe de Lie–Poisson G est son algèbre de Lie g accompagnée d’unestructure supplémentaire : l’espace dual g∗ hérite d’une structure d’algèbre de Lie, satisfaisant une condition decompatibilité avec le crochet de Lie sur g (cf. par exemple [4,5]). Une telle paire d’algèbres de Lie (g,g∗) s’appelleune bialgèbre de Lie. Réciproquement, toute bialgèbre de Lie (g,g∗) est la bialgèbre de Lie d’un groupe de Lie–Poisson G connexe et simplement connexe [4].E-mail address: cortiz@impa.br.1 This research was supported by Capes-Brazil through “Programa de Estudantes-Convênio de Pós-Graduacão PEC-PG” by Capes-Brazil.1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.10.0031280 C. Ortiz / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1279–1282Une structure de Poisson peut être vue comme un cas particulier d’une structure géométrique plus générale : lastructure de Dirac. La notion de structure de Dirac a été introduite par Courant et Weinstein dans le cadre des systèmesméchaniques [2,3]. Les structures de Dirac incluent les structures pré-symplectiques et les structures de Poisson, demême que les feuilletages réguliers. Si (G,π) est un groupe de Lie–Poisson, la multiplicativité de π est équivalenteau fait que l’application fibrée induite π :T ∗G → T G soit un morphisme de groupoïdes. Dans ces conditions, lastructure de Dirac associée Lπ = graph(π) ⊆ T G ⊕ T ∗G est un sous-groupoïde de Lie.Dans cet exposé, nous étudions les groupes de Lie G équipés d’une structure de Dirac L multiplicative dans le sensoù L ⊆ T G ⊕ T ∗G est un sous-groupoïde. Nous appelerons un tel couple (G,L) un groupe de Lie–Dirac.Cet exposé est organisé de la manière suivante : dans le deuxième paragraphe, nous étudions les feuilletagesmultiplicatifs sur un groupe de Lie G, c’est-à-dire les structures de Dirac multiplicatives induites par des distributionsrégulières involutives sur T G. Étant donné un sous-groupe de Lie connexe H d’un groupe de Lie G, il existe unfeuilletage canonique de G, dont les feuilles sont les classes à droite de H . Nous montrons que ce feuilletage estmultiplicatif si et seulement si H est un sous-groupe normal de G.Dans le troisième paragraphe, nous étudions le feuilletage caractéristique d’une structure de Dirac multiplicative.Nous montrons que le noyau d’une structure de Dirac multiplicative sur un groupe de Lie G est une distributionrégulière involutive, que s’intègre comme un feuilletage régulier et multilicatif de G. À l’aide de la description desfeuilletages multiplicatifs donnés au paragraphe 2, nous déduisons que le feuilletage caractéristique d’un groupe deLie–Dirac (G,L) est simple et que l’espace des feuilles est un groupe de Lie–Poisson. C’est-à-dire que les groupesde Lie–Dirac, après un quotient canonique deviennent des groupes de Lie–Poisson. Enfin, nous appliquons ce résultatainsi que la correspondance de Drinfeld entre les groupes de Lie–Poisson et les bialgèbres de Lie pour obtenir laversion infinitésimale des groupes de Lie–Dirac.1. IntroductionA Poisson–Lie group is a Lie group G with a multiplicative Poisson structure π ∈ Γ (Λ2T G), that is, the multi-plication map m :G × G → G is a Poisson map [4,5,7]. Equivalently, the bundle map π :T ∗G → T G is a groupoidmorphism, where the tangent and cotangent bundles have the groupoid structures induced by G (see e.g. [6]). There-fore, the multiplicativity property of the Poisson bivector π is equivalent to saying that the associated Dirac structure[2,3] Lπ = graph(π) is a Lie subgroupoid of the direct-sum V B-groupoid TG = T G ⊕ T ∗G. This motivates thefollowing definition: a Dirac structure L on a Lie group G is multiplicative if L ⊆ TG is a subgroupoid. We refer toa Lie group equipped with a multiplicative Dirac structure as a Dirac–Lie group.Clearly Poisson–Lie groups are particular examples of Dirac–Lie groups. On the other extreme, one can checkthat there are no interesting multiplicative 2-forms on Lie groups – the only one is the zero 2-form. Another class ofexamples is obtained as follows: Let p :G1 → G2 be a homomorphism of Lie groups which is a surjective submersion.If π is a multiplicative Poisson structure on G2, then its pull back (in the sense of Dirac structures [1]) turns out tobe a multiplicative Dirac structure on G1, whose pre-symplectic leaves are the inverse images by p of the symplecticleaves of G2, and whose characteristic foliation is given by the fibres of the submersion p. Our main observation inthis note is that, modulo a regularity condition, all multiplicative Dirac structures on Lie groups are of this form.2. Preliminaries2.1. Multiplicative foliationsLet G be a Lie group. If F ⊆ T G is a regular integrable distribution, one can check that the corresponding Diracstructure LF = F ⊕ Ann(F ) is multiplicative if and only if F ⊆ T G is a Lie subgroup, where T G has the natural Liegroup structure induced from G.Proposition 2.1. Let F be the foliation integrating a multiplicative distribution F ⊆ T G. The following holds:(i) The leaf through the identity Fe ⊆ G is a normal Lie subgroup.(ii) The foliation F is given by cosets of Fe.Proof. Since F ⊆ T G is a subgroup, it is closed under multiplication in T G, that is dm(g,h)(Xg,Xh) = dRh(g)Xg +dLg(h)Xh ∈ Fgh for every Xg,Xh ∈ F . In particular, for Xh = 0 we see that F is right invariant, i.e. dRh(g)Xg ∈ Fgh.C. Ortiz / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1279–1282 1281Similarly we obtain left invariance of F : dLg(h)Xh ∈ Fgh. This says that the distribution at each g ∈ G is given byFg = dLg(e)Fe = dRg(e)Fe. (1)Consider now Fe , the leaf of F through the identity e ∈ G. For every a, b ∈ Fe there exist paths a(t), b(t) ∈ G,t ∈ [0,1], tangent to the distribution F , joining the identity e ∈ G to a and b, respectively. We want to prove thatc = ab ∈ Fe. For this, take the path c(t) = a(t)b(t), which joins the identity to c = ab. The path c(t) is tangent to thedistribution F : indeed, the bi-invariance of F implies thatc′(t) = dRb(t)(a(t))a′(t) + dLa(t)(b(t))b′(t) ∈ Fc(t),since a′(t) ∈ Fa(t) and b′(t) ∈ Fb(t). This shows that c ∈ Fe. A similar computation shows that Fe is closed by theinversion map. Therefore the leaf through the identity is a subgroup of G. Moreover, it follows from (1) that the Liealgebra of Fe is Ad-invariant, which is equivalent to Fe being a normal subgroup. The assertion in (ii) follows fromthe bi-invariance in (1). 2.2. Functorial properties of multiplicative Dirac structuresLet M be a smooth manifold. The generalized tangent bundle of M is the direct-sum vector bundle TM = T M ⊕T ∗M . Given a smooth map ϕ :M1 → M2 and x ∈ M1, we say that the elements η = (X,α) ∈ (TM1)x and ξ = (Y,β) ∈(TM2)ϕ(x) are ϕ-related if Y = dϕ(X) and α = dϕ∗β . For a Lie group G with Lie algebra g, T G is a Lie group, T ∗Gis a Lie groupoid over g∗ and we consider the groupoid TG = T G ⊕ T ∗G; we denote the groupoid multiplication inTG by ξ ∗ ξ ′, when ξ, ξ ′ are composable.Lemma 2.1. Let ϕ :G1 → G2 be a homomorphism of Lie groups, which is a surjective submersion. If ξϕ(g), ξ ′ϕ(h) ∈TG2 are ϕ-related to ηg, η′h ∈ TG1, respectively, then ξϕ(g), ξ ′ϕ(h) are composable if and only if ηg, η′h are composable.Moreover, ξϕ(g) ∗ ξ ′ϕ(h) ∈ TG2 is ϕ-related to ηg ∗ η′h ∈ TG1.Proof. The pull back bundles ϕ∗(T G2), ϕ∗(T ∗G2) have natural groupoid structures in such a way that dϕ :T G1 →ϕ∗(T G2) and dϕ∗ :ϕ∗(T ∗G2) → T ∗G1 are morphisms of groupoids. The statements follow from this fact and a directcomputation using that ϕ is a surjective submersion. Corollary 2.2. Let ϕ :G1 → G2 be a homomorphism of Lie groups, which is a surjective submersion. Assume thatL1,L2 are Dirac structures on G1,G2, respectively. If ϕ is a forward Dirac map and L1 is multiplicative, then L2 ismultiplicative. Also, if ϕ is a backward Dirac map and L2 is multiplicative, then L1 is multiplicative.Proof. Recall that ϕ is a forward Dirac map if and only if L2 is the bundle of all ϕ-related elements to elements in L1.A backward Dirac map is defined in a similar way, see [1]. The statement follows from Lemma 2.1. 3. The main resultLet L be a Dirac structure on a smooth manifold M . Let us denote by pT the canonical projection of T M ⊕ T ∗Mon T M . The manifold M carries a singular foliation tangent to the generalized distribution pT (L) ⊆ T M , and eachleaf of this foliation inherits a canonical pre-symplectic structure whose kernel is ker(L) = L∩T M . If the distributionker(L) has constant rank, then it is integrable. If the corresponding foliation K is simple, then the leaf space M/K hasa natural Poisson structure obtained by identifying functions on the quotient with admissible functions on M , see [2]for details. The foliation K is called the characteristic foliation of L.In the special case of Dirac–Lie groups our main result is the following:Theorem 3.1. Let G be a Lie group with a multiplicative Dirac structure L ⊆ T G ⊕ T ∗G. Then:(i) The kernel of L is a multiplicative integrable distribution, and the leaves of the characteristic foliation K arecosets of the normal Lie subgroup Ke ⊆ G.1282 C. Ortiz / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1279–1282(ii) If Ke is closed,2 then the leaf space G/K is smooth and the induced Poisson structure π is multiplicative (i.e.,G/K becomes a Poisson–Lie group). Moreover, L is the pull back of π by the quotient map G → G/K.Proof. Since L is multiplicative, we have that ker(L) = L ∩ T G ⊆ T G is a subgroup, hence (1) implies that ker(L)has constant rank. In particular it defines an involutive distribution, whose leaves are given by cosets of the normal Liesubgroup K = Ke (the leaf through the identity) by Proposition 2.1. If K is closed, then G/K is a Lie group and theprojection G → G/K is a surjective submersion which is both a forward and backward Dirac map [1], where G/K isequipped with the natural Poisson structure induced by L. The multiplicativity property of this Poisson structure is aconsequence of Corollary 2.2. Combining Theorem 3.1 and Drinfeld’s correspondence between Poisson–Lie groups and Lie bialgebras [4], weobtain the infinitesimal counterpart of Dirac–Lie groups.Corollary 3.2. Let G be a Lie group with Lie algebra g. If G is equipped with a multiplicative Dirac structure L, thenk = ker(L)e is an ideal in g and the quotient g/k inherits the structure of a Lie bialgebra.Proof. The multiplicativity of the characteristic distribution implies that k ⊆ g is an ideal. Now consider the con-nected and simply connected Lie group T integrating the quotient Lie algebra g/k. The canonical projection g → g/kintegrates to a homomorphism of Lie groups φ : G̃ → T , where G̃ denotes the universal covering of G. The subgroupH = ker(φ) is closed and normal in G̃, therefore the connected component of the identity H0 is closed and normal aswell and the quotient group G̃/H0 inherits a Poisson–Lie structure. Since G̃/H is locally diffeomorphic to G̃/H0, theLie algebra g/k inherits a Lie bialgebra structure. In the situation of Corollary 3.2 we say that (G,L) is an integration of the infinitesimal data (g, k), where k ⊆ g isideal and g/k is a Lie bialgebra.Corollary 3.3. If G is connected and simply connected and k ⊆ g is an ideal such that g/k is a Lie bialgebra, thenthere is a unique multiplicative Dirac structure on G integrating (g, k).Proof. Let T be the connected and simply connected Lie group integrating g/k. Consider the homomorphismφ :G → T and H ⊆ G as in the proof of Corollary 3.2. The quotient group G/H ∼= T has a multiplicative Pois-son structure πT integrating the Lie bialgebra g/k. Since φ is a surjective submersion, we induce a multiplicativeDirac structure L on G according to Corollary 2.2. This shows that (G,L) is an integration of (g, k). AcknowledgementsI would like to thank Henrique Bursztyn for helpful discussion and valuable comments. I thank also Marius Crainicfor helpful suggestions. Finally I thank the Department of Mathematics at Utrecht University for hospitality while partof this work was carried out.References[1] H. Bursztyn, O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble) 53 (2003) 309–337.[2] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990) 631–661.[3] T. Courant, A. Weinstein, Beyond Poisson structures, in: Action hamiltonnienes de groupes. Troisieme théorème the Lie, Lyon, 1986, Hermann,Paris, 1988, pp. 39–49.[4] V. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and geometric meaning of the classical Yang–Baxter equations, Soviet Math.Dokl. 27 (1) (1983) 68–71.[5] J.H. Lu, A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geometry 31 (1990) 501–526.[6] K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Notes Ser., vol. 213, 2005.[7] M. Semenov-Tian-Shansky, Dressing transformation and Poisson–Lie group actions, Publ. RIMS, Kyoto University 21 (1985) 1237–1260.2 Notice that not always the characteristic leaf through the identity Ke ⊆ G is a closed subgroup. Indeed, if G is the torus and K ⊆ G is a densegeodesic, then the Dirac structure associated to the foliation of G by cosets of K is a multiplicative Dirac structure whose characteristic leaf throughthe identity is dense.

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