The graded poissonlie structures derived before for gln and sln are shown to be the special cases of this bracket. We study poisson and liepoisson structures on the diffeomorphism groups with a smooth metric spray in connection with dynamics of nonlinear pdes. One interesting, and somewhat surprising, fact is that any poisson manifold is the codomainimage of a surjective, submersive poisson map from a symplectic manifold. Some remarks on invariant poisson quasinijenhuis structures. The poisson lie groups g, n and g, n are said to be dual to each other. The formula is universal and can be applied to any matrix lie group. Vezzosi draft, may 2015 abstract this is the sequel of ptvv in which we introduce and discuss the notion of nshifted poisson. Henrique bursztyn, marius crainic, dirac structures, momentum maps and quasi poisson manifolds. We study the poisson lie structures on the group su2,r. The lie algebra of a poissonlie group turns out to be a lie bialgebra with cobracket the derivative of p. Sg0412082, noncommutative geometry and representation theory in mathematical physics, 1234, contemporary mathematics 391.
On the other hand, affine poisson structures on a given lie group. Definition of multiplicative poisson structures on lie groups g poisson lie groups. Said in another manner, it is a lie algebra structure on the vector space of smooth functions on such that. Liepoisson structures, poissonlie groups, hamiltonian systems, bihamiltonian systems, completely integrable systems, integrable deformations, coalgebras some figures may appear in colour only in the online journal 1.
Linear poisson structures and lie algebras springerlink. Pdf on the structure of infinitesimal automorphisms of. On deformation quantization of poissonlie groups and. We show that all these structures are linearizable in. Banach poisson lie groups and related structures 5 1. This note is based on the observation made in that poissonlie groups, and many related poisson manifolds, can be realized as moduli spaces of flat connections on surfaces with boundary and marked points, which makes the problem of their deformation quantization rather straightforward. Triangular poisson structures on lie groups and symplectic. Thus the construction of the kirillov symplectic form is generalized for liepoisson groups. Mixed product poisson structures associated to poisson lie. In this thesis a thorough introduction to poissonlie groups is given. For instance, with their standard symplectic structures, there do not exist poisson maps, whereas symplectic maps abound.
Poisson structures for pdes associated with diffeomorphism groups. There are two main motivations for studying poisson lie groups. Invariant poissonnijenhuis structures on lie groups and classification zohreh ravanpak, adel rezaeiaghdam, and ghorbanali haghighatdoost abstract. We study rightinvariant respectively, leftinvariant poissonnijenhuis structures pn on a lie group g and introduce their infinitesimal counterpart, the socalled rn structures on the corresponding lie algebra we show that rn structures can. More precisely, a poisson lie group is a lie group with a poisson structure such that its group multiplication is a poisson morphism. We would like to thank the organizers of the ictp program on poisson geometry for organizing such a wonderful event. Aspects of the poisson structures to be investigated include calculations. Xiaomeng xu stokes phenomenon and poisson lie groups. Ane poisson structures on lie groups also serve as the target poisson structures for a not necessarily equivariant momentum mapping. Invariant poissonnijenhuis structures on lie groups and. We let kbe a eld we care mostly about real or complex numbers. Drinfeld lie bialgebra structures on simple lie algebra. Banach poissonlie groups and related structures 5 1. Stokes matrices, poisson lie groups and frobenius manifolds 483 of g via the pairing.
A poisson structure on a manifold p is a lie algebra operation, on. Volume 5, number 2, april 1992 american mathematical. A lie bialgebra is a lie algebra which is at the same time a lie coalgebra, the algebra and coalgebra structures satisfying some compatibility relation. Problems concerning singularities of poisson structures 37 3. On poissonlie structure on the external algebra of the. In each one of these contexts, it turns out that the poisson structure is not a theoretical. The corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra of. Quantum spaces give rise to poisson structures as semiclassical limits. The lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered. Mixed product poisson structures associated to poisson lie groups and lie bialgebras article pdf available in international mathematics research notices 201719.
Mixed product poisson structures associated to poisson lie groups and lie bialgebras article pdf available in international mathematics research notices 201719 april 2015 with 82 reads. In other words, this is a pair of dual vector spaces with compatible lie algebra structures. The prolate spheroidal phenomenon and bispectral algebras of ranks 1 and 2 with alberto grunbaum, latest file pdf, ps. Together with symplectic manifolds, considered in the previous chapter, lie algebras provide the first examples of poisson manifolds. Poisson structures appear in a large variety of contexts, ranging from string theory, classicalquantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory.
Volume 5, number 2, april 1992 american mathematical society. Conversely, two compatible rmatrices from which one is. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. We show that each triangular poisson lie group can be decomposed into poisson submanifolds each of which is a quotient of a symplectic manifold. Symplectic structures associated to liepoisson groups. For the natural poisson brackets the symplectic leaves in these manifolds are classified, and the corresponding symplectic forms are described. Apr 26, 2015 mixed product poisson structures associated to poisson lie groups and lie bialgebras article pdf available in international mathematics research notices 201719 april 2015 with 82 reads. A poisson group structure on a lie g r oup g is g iv en by specifying a lie bia l gebra str ucture on its lie algebra g. These are objects whose algebra of functions can be deformed into a hopf algebra. The liepoisson structure of the euler equations of an ideal. We prove a reduction theorem explaining how different parts of. Here, is the lie algebra of the lie group and the linear mapping is defined to be.
We study rightinvariant respectively, left invariant poisson quasinijenhuis structures on a lie group g and introduce their infinitesimal counterpart, the socalled rqn structures on the corresponding lie algebra. Now lets add some structure to our categories and see how it reflects on our algebra. Vorobjev geometric data and transversal structures 19 chapter 2. A lie group with a poisson structure which is compatible with the group multiplication, i. Poisson structures for pdes associated with diffeomorphism.
Introduction a poisson group structure on a lie group g is given by specifying a lie bialgebra structure on its lie algebra g. A poisson lie group often written poissonlie group or lie poisson group g g is a lie group and a poisson manifolds, the two structures being compatible such that the group product is a smooth map of poisson manifolds where g. The marsdenweinsteinmeyer symplectic reduction technique is. In this chapter we discuss poissonlie groups and their infinitesimal counterparts lie bialgebras. Introduction to poisson lie groups german stefanich tannaka duality our starting point will be the tannaka duality, which is a dictionary between two worlds, one which we could call algebra, and the other one representation theory or noncommutative geometry. I quantum groups are quantizations of pl structures on lie groups. Lie algebras of lie groups for the algebraic theory of lie algebras, see 17bxx 53c15. Invariant structures on the lie group can be identi. The results are illustrated in detail for the generalized jordanian poisson structures on sln. We study poisson and lie poisson structures on the diffeomorphism groups with a smooth metric spray in connection with dynamics of nonlinear pdes. M is a poisson algebra if and only if m is a poisson module. Such a lie bialgebra structure is said to be triangular if the cocommutator.
Poissonlie groups i pl groups are quadratic poissonhopf algebras on group manifolds that are invariant under the group product the coproduct map. A poissonlie group is a lie group g which is equipped with a poisson structure. Poissonlie structure on the tangent bundle of a poissonlie group and poisson action lifting boumaiza, mohamed and zaalani, nadhem, journal of geometry and symmetry in physics, 2005 infinite dimentional lie groups with applications to mathematical physics schmid, rudolf, journal of geometry and symmetry in physics, 2004. One may then think of algebras as just a way of presenting categories. Poisson structures camille laurentgengoux springer. Alekseevmeinrenken for coboundary poisson lie groups, gelfandzeitlin integrable systems alekseevmeinrenken, stokes phenomenon boalch for complex reductive standard poisson lie groups, dynamical rmatrices and quasihopf algebra enriquezetingofmarshall in a formal setting. Liepoisson structure on some poisson lie groups american. Poisson structures on double lie groups sciencedirect. Poissonlie groups are nontrivial combinations of lie groups and poisson manifolds.
In particular, we provide a precise analytic sense in which the time t map for the euler equations of an ideal fluid in a region of rn or on a smooth compact nmanifold with a boundary is a poisson map relative to the lie poisson bracket. Victor ginzburg, alan weinstein, lie poisson structure on some poisson lie groups, journal of the ams, volume 5, number 2 1992. Poisson lie groups are nontrivial combinations of lie groups and poisson manifolds. Multiplicative and ane poisson structures on lie groups. The class of elementary lie algebras is defined and the poisson and symplectic structures for them are described. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra. G has been given the structure of a product poisson manifold. Pdf triangular poisson structures on lie groups and. On deformation quantization of poissonlie groups and moduli. The symplectic lie groupoid lie integrating lie poisson structures is discussed as example 4.
This paper is the first of a series of papers devoted to a detailed study of some naturally defined holomorphic poisson structures on products of flag varieties and related spaces of complex semisimple lie groups see sections 1. In these examples, the information at the level of poissonlie groups or lie bialgebras plays a key role in constructing the quantum groups and their structures. A lie bialgebra is a lie algebra which is at the same time a lie coalgebra, the algebra and coalgebra structures. Basic constructions alie bialgebrais a lie algebra g together with a compatiblelie algebra structure on g. In particular, we provide a precise analytic sense in which the time t map for the euler equations of an ideal fluid in a region of rn or on a smooth compact nmanifold with a boundary is a poisson map relative to the liepoisson bracket. This gives a new class of solutions of the classical yangbaxter equation. Manin triples in the in nitedimensional setting 5 1. Liepoisson structure on some poisson lie groups semantic scholar.
Pdf mixed product poisson structures associated to poisson. We study rightinvariant respectively, left invariant poisson quasinijenhuis structures on a lie group g and introduce their infinitesimal counterpart, the socalled rqn structures on the corresponding lie algebra we investigate the procedure of the classification of such structures on the lie algebras and then for clarity of our results we classify, up to a natural equivalence, all r. Poisson lie structure on the tangent bundle of a poisson lie group and poisson action lifting boumaiza, mohamed and zaalani, nadhem, journal of geometry and symmetry in physics, 2005 infinite dimentional lie groups with applications to mathematical physics schmid, rudolf, journal of geometry and symmetry in physics, 2004. Quasi,twisted,andallthat inpoissongeometryandliealgebroidtheory. Poisson lie groups, lie bialgebras, lie poisson spaces, schatten ideals. A poissonlie group is a lie group g equipped with a poisson bracket for which the group multiplication. Poissonlie groups uvafnwi universiteit van amsterdam. Quantum groups give rise to poisson structures on lie groups and related spaces. Triangular poisson structures on lie groups and symplectic reduction with tim hodges, pdf, ps, math. Lie group is a lie group gwith a poisson structure.
General geometric structures on manifolds almost complex, almost product structures, etc. We study rightinvariant respectively, leftinvariant poissonnijenhuis structures pn on a lie group g and introduce their infinitesimal counterpart, the socalled rn structures on the corresponding lie algebra we show that rn structures can be used to find compatible solutions of the classical yangbaxter equation cybe. The standard rmatrix in a manin decomposition then gives rise to several poisson structures on the correponding double group, which is investigated in great detail. The notion of liepoisson structures was originally found by sophus lie and then rediscovered by felix berezin and by alexander kirillov, bertram kostant and. Furthermore, he is committed to teaching and set up several mathematics projects with local high schools. Laurentgengoux research focus lies on poisson geometry, lie groups and integrable systems. Pdf cluster structures on simple complex lie groups and. Under addition a liepoisson manifold becomes a poisson lie group, see there for more related concepts. We calculate all poisson lie structures on su2,r through the correspondence with lie bialgebra structures on its lie algebra su2,r. Dirac structures, momentum maps and quasipoisson manifolds.
Introduction to poissonlie groups german stefanich tannaka duality. In recent work, lu and ratiu lr used socalled standard poisson structures on a compact semisimple lie group k and on its poisson dual k in order to give a. Stokes matrices, poisson lie groups and frobenius manifolds. Pdf mixed product poisson structures associated to. Poissonlie groups, lie bialgebras, liepoisson spaces, schatten ideals. We study natural cluster structures in the rings of regular functions on simple complex lie groups and poissonlie structures compatible with these cluster structures. This poisson bracket is also called the \liepoisson bracket, to emphasize the fact that it actually was already known to lie. G define an equivalence relation between lie bialgebra structures over the lie algebra g of g. Poisson algebra the corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra of. In some cases, the compatible poisson brackets on the poisson lie groups are shown to be of our special type, in which case we can apply the result of this.
The purpose of this paper is to describe a new class of poisson and symplectic structures on lie algebras. In geometry, a poisson structure on a smooth manifold is a lie bracket. Multiplicative and affine poisson structures on lie groups hku. The liepoisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered.
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