Poisson lie group pdf

Here, is the lie algebra of the lie group and the linear mapping is defined to be. Although this is the obvious definition, neither left translations nor right translations are poisson maps. Poisson lie group, tangent lie bialgebra of a poisson lie group, exponential map for lie bialgebras. Pdf poisson lie groups and their relation to quantum groups. Examples of poisson brackets department of mathematics. Semenovtianshansky to understand the hamiltonian structure.

Ergebnissedermathematik volume51 undihrergrenzgebiete. A poissonlie group is a lie group g which is equipped with a poisson structure. Lie groups which are endowed with multiplicative poisson structures are called poissonlie groups and liepoisson structures on the dual space of a lie algebra g are examples of abelian poissonlie groups for more details see, for instance, 23. Therefore, pl groups are poissonhopf algebras endowed with a coproduct map. Multiplicative and ane poisson structures on lie groups. Poisson lie groups are instances of poisson coalgebras for which the comultiplication map is given by the group law, and the quantization of the former are the socalled quantum groups, which are the underlying symmetries of many relevant quantum integrable models see, for instance 8, 10. Central extensions of lie bialgebras and poissonlie groups. This is achieved with the help of the exponential map, which is a local di. This is achieved by finding explicit darboux coordinates and then using a phase space path integral.

We study the poissonlie structures on the group su2,r. We let kbe a eld we care mostly about real or complex numbers. In this way we obtain the unitary representations of uqsu2. This textbook introduces modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. Theorem the cotangent lift action is qhamiltonian, and the reduced symplectic qmanifold is t1mg. On the vector space g dual to the lie algebra g of gthere is a natural poisson structure. Heisenberg lie group h 4 has poissonlie symmetry only when the dual lie group is a 2.

Recall that a poisson lie group is a poisson manifold in the lie category. Reduced riemannian poisson manifolds and riemannian. The marsdenweinsteinmeyer symplectic reduction technique is. Stokes phenomenon and poisson lie groups xiaomeng xu course 18. 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.

We will see that the failure in this equality can be controlled in much the same way. Goldman department ofmathematics, massachusetts institute oftechnology, cambridge, ma029, usa in7itwas shownthatifn is thefundamental groupofa closed orientedsurface s and gis lie groupsatisfying very general conditions, then the space homn,gg. Let mbe a poisson manifold and ga lie group acting on m by poisson automorphisms. Invariant functions on lie groups and hamiltonian flows ofsurface group representations william m. Quazitriangular lie bialgebras, factorizable lie bialgebras. Under addition a liepoisson manifold becomes a poisson lie group, see there for more related concepts.

In particular, we shall develop the discrete analogue of euler. Said in another manner, it is a lie algebra structure on the vector space of smooth functions on such that. The notion of liepoisson structures was originally found by sophus lie and then rediscovered by felix berezin and by alexander kirillov, bertram kostant and. Central extensions of poissonlie groups in the sequel, let g be a poissonlie group 1,3,5 and a a 1dimensional abelian poissonlie group. The notion of poisson lie group sometimes called poisson drinfeld group was first introduced by v. A riemannpoisson lie group is a lie group endowed with a left invariant riemannian metric and a left invariant poisson tensor which are compatible in the sense introduced in.

A lie group with a poisson structure which is compatible with the group multiplication, i. Scaling of poisson spheres and compact lie groups sheu, albert jeuliang, asian journal of mathematics, 2012. Integrability of poissonlie group actions 3 in general, smg 6smg. Compatibility conditions on the reduced poissonlie group. The liepoisson structure of the euler equations of an. Integrability of poissonlie group actions springerlink. Lie groups are named after norwegian mathematician sophus lie, who laid the foundations of the theory of continuous transformation groups. In geometry, a poisson structure on a smooth manifold is a lie bracket. Let us look at the relation between a lie group and its lie algebra as an example.

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. Poisson lie symmetry and dbranes in wzw model on the. Lie algebras of vector fields and related super algebras 22e65. Dressing orbits and a quantum heisenberg group algebra kahng, byungjay, illinois journal of mathematics, 2004. In terms of linear coordinates e i and fk ij the structure constants of the group it has the form fe i. In this chapter we discuss poissonlie groups and their infinitesimal counterparts lie bialgebras. Path integral quantization of the symplectic leaves of the. Invariant functions on lie groups and hamiltonian flows. G,h,ig be a poisson liegroup equipped with a left invariant riemannian metric and h a normal and closed coisotropic subgroup of g. Pdf on the structure of infinitesimal automorphisms of. Triangular poisson structures on lie groups and symplectic reduction timothy j.

We calculate all poissonlie structures on su2,r through the correspondence with lie bialgebra structures on its lie algebra su2,r. Its symplectic leaves are the orbits of the coadjoint. This means that nontrivial poissonlie structure is never symplectic, not even of constant rank. Lectures on calogeromoser systems pavel etingof to my mother yelena etingof on her 75th birthday, with admiration. We therefore expect a correspondence between points in g and irreducible ogmodules. Mixed product poisson structures associated to poisson lie. The strategy is to analyze and collect enough information from a poissonlie group, and using it to carry out a co. The liepoisson structure of the euler equations of an ideal fluid. Poissonlie groups, bihamiltonian systems and integrable. In particular, it explains the dynamics of rotating, spinning and rolling rigid bodies from a. Poisson groupoids were introduced by alan weinstein as generalizations of both poisson lie groups and symplectic groupoids. Pdf lie, symplectic and poisson groupoids and their lie. For an action of a poissonlie group g on a poisson manifold m, we find an explicit description of the lifted hamiltonian action on the symplectic groupoid. Example if m is a poisson manifold and g is a poissonlie group with a free and proper poisson action on m, then the poisson quotient mg.

They were introduced by drinfeld 1, 2 in connection with quantum groups, which are just hopf algebra quantizations of pl structures see 112 and references therein. In this way we obtain the unitary representations of. We present here the main definitions and first properties relative to groupoids, lie groupoids, lie algebroids, symplectic and poisson groupoids and their lie algebroids. We show that each triangular poisson lie group can be decomposed into poisson submanifolds each of which is a quotient of a symplectic manifold. Poisson lie groups university of california, berkeley. We give a oneline proof of this fact in appendix 2, using the poisson calculus. A lie bialgebra is a lie algebra which is at the same time a lie coalgebra, the algebra and coalgebra structures. Poissonlie pl groups are poisson structures on a lie group g for which the group multiplication is a poisson map. The feynman path integral is used to quantize the symplectic leaves of the poissonlie group su2. G, whose schouten bracket does not vanish so that it is. A lie group approach to solve the fractional poisson equation. Poisson algebra the corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra of.

Indeed, ogmodules are sheaves on g, and one can think of an arbitrary sheaf as a sum of skyscrapper sheaves, which in turn correspond to points in g. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of poisson structures and of the usually mathematical contexts in which they appear, and the final part is devoted to the two main applications of poisson structures integrable systems and deformation quantization. Examples of poisson brackets weinstein symposium ihp, july 18 20, 20 jianghua lu department of mathematics the university of hong kong july 18, 20. Lee is a group whose elements are organized continuously and smoothly, as opposed to discrete groups, where the elements are separatedthis makes lie groups differentiable manifolds. We introduce and study some mixed product poisson structures on product manifolds associated to poisson lie groups and lie bialgebras. Noncoboundary poissonlie structures on the book group. A lie group equipped with a multiplicative poisson structure is called a poisson lie group. We show that all these structures are linearizable in.