Poisson lie algebra book

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. In mathematics, a poissonlie group is a poisson manifold that is also a lie group, with the. The corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra. Why does my book consider moment of inertia as a scalar when it is a tensor.

Cluster algebras and poisson geometry michael gekhtman. In this paper, the notions of vertex lie algebra conformal algebra and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. This poisson structure on g will be of central importance from now on. Poisson structures on moduli spaces of representations. A lie group with a poisson structure which is compatible with the group multiplication, i. Lie algebra of classical observables under poisson bracket. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a lie bialgebra on a coboundary. It is well known that then a,0, is a poisson algebra which is referred to as the classical limit of the deformation dri, section 2. The generalized weyl poisson algebras and their poisson.

All classical observables form a lie algebra under the poisson bracket. Similarly, one may look for deformations of a zinbiel algebra into dendriform algebras, and wonder. Poisson algebras appear naturally in hamiltonian mechanics, and are also central in the study of quantum groups. They are also called poissonhopf algebras, and are the lie algebra of a poissonlie group. In mathematics, a poisson algebra is an associative algebra together with a lie bracket that. Poisson actions of g then correspond to hamiltonian actions of go together with their momentum maps.

In mathematics, a poisson algebra is an associative algebra together with a lie bracket that also satisfies leibnizs law. Vertex lie algebras, vertex poisson algebras and vertex. Poisson structure an overview sciencedirect topics. 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. The lie algebra of a poisson lie group turns out to be a lie bialgebra with cobracket the derivative of p. Alie bialgebrais a lie algebra g together with a compatiblelie algebra structure on g. Here we return to our motivating discussion of poisson manifolds, especially reasons 2 and 3 of.

In this chapter we discuss poissonlie groups and their infinitesimal. Now we can put the above two stories on an equal footing. This paper studies certain relations among vertex algebras, vertex lie algebras and vertex poisson algebras. Actions of lie groups and lie algebras on symplectic and poisson. The notions of vertex lie algebra and vertex poisson algebra are presented and connections among vertex lie algebras, vertex poisson algebras and vertex algebras are discussed. The global poisson webinar aims at presenting talks in the field of poisson geometry, understood in a very broad sense. The question of reconstructing the deformation a h from a poisson algebra is 2. Nathan jacobson, presently henry ford ii professor of mathematics at yale university, is a wellknown authority in the field of abstract algebra. Needless to say, the poisson bracket preservation property has the same proof as 2.

Poisson geometry 225 this theorem unifies the two examples of complete poisson maps which were classified in section 6. The dual equation of a finitedimensional lie algebra equation. The notions of ve rtex lie algebra and verte x poisson algebra are presen ted and connect ions among vertex lie algebras, vertex poisson algebras and v ertex algebras are discussed. Poisson lie groups and lie bialgebras has been proved by drinfeld 6. When m is a lie poisson space g, g is tgo, where go is the connected and simply connected lie group whose lie algebra is g. Lie poisson space b, where b is the threedimensional book algebra so. This means that on linear functions the liepoisson bracket is simply the. Poisson lie pl groups are poisson structures on a lie group g for which the group multiplication is a poisson map. Twisting poisson algebras, copoisson algebras and quantization. They were introduced by drinfeld 1, 2 in connection with quantum groups, which are just hopf algebra quantizations of pl structures see 1 12 and references therein. In mathematics, a poissonlie group is a poisson manifold that is also a lie group, with the group multiplication being compatible with the poisson algebra structure on the manifold. Examples of poisson brackets department of mathematics. This chapter can be viewed as a primer on the theory of cluster algebras. The main goal of this paper is to extend this theory to direct limit lie algebras.

Check out the new look and enjoy easier access to your favorite features. Let g be a nitedimensional lie algebra, with basis and corresponding structure. One of the worlds foremost geometers, alan weinstein has made deep contributions to symplectic and differential geometry, lie theory, mechanics, and related fields. Manifolds with a poisson algebra structure are known as poisson manifolds, of which the symplectic manifolds and the. To quantise functions on a group as a poisson algebra, quantise an appropriate dual. R poisson algebra are particular cases of poisson lie book groups. Complete commutative subalgebras in polynomial poisson. Our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations. Xiaomeng xu stokes phenomenon and poisson lie groups. Quantum algebras as quantizations of dual poissonlie. Noncoboundary poissonlie structures on the book group. The most classical example of a poisson algebra already considered by poisson himself is the algebra of smooth functions on r2nendowed with usual.

Review of the formal deformation quantization of lie poisson structures via transfer of the product on the universal enveloping algebra of the given lie algebra is for instance in. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327. The algebra of a poissonlie group is a lie bialgebra. All possible poisson lie pl structures on the 3d real lie group generated by a dilation and two commuting translations are obtained. Poisson algebra the corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra of. G is a poisson lie group, then the linearization of. Any associative algebra can be seen as a poisson algebra with the trivial lie structure, and any graded lie algebra can be seen as a poisson algebra with the trivial product. Complete commutative subalgebras in polynomial poisson algebras. Written in his honor, the invited papers in this volume reflect the active and vibrant research in these areas and are a tribute to weinsteins ongoing influence. Poisson lie algebra is the lie algebra of the quantomorphism group sch78, rs81. Here, is the lie algebra of the lie group and the linear mapping is defined to be. If one has a lie bialgebra g, the hopf algebra sg is commutative and cocommutative, and has both a poisson bracket. This book is the first devoted to cluster algebras. A poisson algebra is a commutative associative algebra with a unit a equipped with a lie bracket also called a poisson bracket, such that the leibnitz identity holds.

Linear poisson structures and lie algebras springerlink. In geometry, a poisson structure on a smooth manifold is a lie bracket. The clear structure of the book makes it adequate for readers who come across poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of poisson structures. We show that each triangular poisson lie group can be decomposed into poisson submanifolds each of which is a quotient of a symplectic manifold. The starting point of liepoisson theory is the observation that the symplectic leaves of g. By construction, all these poisson structures are quadratic poisson hopf algebras for which the group. Poisson structures mathematical association of america. Triangular poisson structures on lie groups and symplectic reduction timothy j. The proof of the jacobi identity follows from because the lie bracket of vector fields is just their commutator as differential operators the algebra of smooth functions on m, together with the poisson bracket forms a poisson algebra, because it is a lie algebra under the poisson bracket, which additionally satisfies leibnizs rule. Besides the poisson structure on a symplectic manifold, the lie poisson bracket on g, the dual of a lie algebra, is perhaps the most fundamental example of a poisson structure 1999. Liepoisson theory for direct limit lie algebras sciencedirect. Twisting poisson algebras, copoisson algebras and quan tization.

We welcome suggestions of talks for this seminar which should be sent to one of the seminar organizers. Structure classical mechanics quantum mechanics space of states phase space m1 hilbert space h. As an application, we show that the symmetric algebra of a finite dimensional lie algebra, equipped with its natural poisson structure, satisfies the poisson dixmiermoeglin equivalence. Pdf vertex lie algebras, vertex poisson algebras and.

In a more general sense, the poisson bracket is used to define a poisson algebra, of which the algebra of functions on a poisson manifold is a special case. Examples of poisson brackets weinstein symposium ihp, july 18 20, 20 jianghua lu. Vertex algebras and vertex poisson algebras internet archive. Pdf poissonhopf algebra deformations of liehamilton. We show that the two different qdeformed versions of the sl2. Though it presupposes knowledge of linear algebra, it is not overly theoretical and can be readily used for selfstudy. Keywords a generalized weyl poisson algebra a poisson algebra the poisson centre a poisson prime ideal the poisson simplicity. Poisson lie group admits a leftcovariant noncommutative di. Lie algebra, graded respectively, where the grading is shifted by, such that a graded version of the leibniz rule holds.

The full set of poisson lie pl structures on the 3d book group are presented. A graded poisson respectively, gerstenhaber algebra structure on is a graded lie algebra structure cf. Apoisson spaceis a space x with a poisson algebra structure on its algebra of functions. We are also happy to publish dates and access data of other poisson. Their classification is fully performed by relating these pl groups to the corresponding lie bialgebra structures on the corresponding book lie algebra. This is called a poisson bracket, and the data of a commutative algebra with a poisson bracket is called a poisson algebra. Let g lie algebra lie group lie subalgebra linear isomorphism neighborhood norm notation note obtain open subset operator of order poisson algebra polynomial positive constant product integral proof of theorem proposition prove. For each such cluster algebra we construct a large family of clusters and explicit mutations between them.

Although the link with lie groups is important for applications, the present paper is only concerned with lie algebras, so we will be able to handle noncompactaswell ascompactmanifolds x. The lie transformation acting on the poisson bracket of functions gives the same result as the poisson bracket of the lie transformations acting separately on the single functions involved in. Triangular poisson structures on lie groups and symplectic. After presenting the necessary introductory material about poisson geometry and schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in chapter 3. In particular this applies to necklace lie algebra for path algebras of doubled quivers and preprojective algebras. Hopf algebra deformations are merged with a class of lie systems of hamiltonian type, the socalled lie hamilton systems, to devise a novel formalism. The marsdenweinsteinmeyer symplectic reduction technique is. Lie bialgebras, poisson lie groups and dressing transformations. The wellrecognized contributors to this text cover a broad. Pdf twisting poisson algebras, copoisson algebras and. Poissonlie tduality of wzw model via current algebra.