Vertex lie algebras, vertex poisson algebras and vertex. Let g be a nitedimensional lie algebra, with basis and corresponding structure. This book is the first devoted to cluster algebras. 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. Hopf algebra deformations are merged with a class of lie systems of hamiltonian type, the socalled lie hamilton systems, to devise a novel formalism. 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. This may be thought of, roughly, as a commutative algebra together with a rst order jet of a deformation. Poisson structures on moduli spaces of representations. Quantum algebras as quantizations of dual poissonlie.
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. Pdf twisting poisson algebras, copoisson algebras and. 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. 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 main goal of this paper is to extend this theory to direct limit lie algebras. In this chapter we discuss poissonlie groups and their infinitesimal. The dual to a finitedimensional lie algebra has a natural structure of a poisson manifold, the liepoisson structure. Why does my book consider moment of inertia as a scalar when it is a tensor. 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. Complete commutative subalgebras in polynomial poisson. Twisting poisson algebras, copoisson algebras and quantization. Poissonlie tduality of wzw model via current algebra.
By construction, all these poisson structures are quadratic poisson hopf algebras for which the group. Similarly, one may look for deformations of a zinbiel algebra into dendriform algebras, and wonder. A graded poisson respectively, gerstenhaber algebra structure on is a graded lie algebra structure cf. The lie algebra of a poisson lie group turns out to be a lie bialgebra with cobracket the derivative of p. This is called a poisson bracket, and the data of a commutative algebra with a poisson bracket is called a poisson algebra. Actions of lie groups and lie algebras on symplectic and poisson. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a lie bialgebra on a coboundary. All classical observables form a lie algebra under the poisson bracket. Vertex algebras and vertex poisson algebras internet archive. This means that on linear functions the liepoisson bracket is simply the. 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. The global poisson webinar aims at presenting talks in the field of poisson geometry, understood in a very broad sense.
G is a poisson lie group, then the linearization of. In mathematics, a poissonlie group is a poisson manifold that is also a lie group, with the. 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. Nathan jacobson, presently henry ford ii professor of mathematics at yale university, is a wellknown authority in the field of abstract algebra. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327. The corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra. This chapter can be viewed as a primer on the theory of cluster algebras. Cluster algebras and poisson geometry michael gekhtman. Poisson algebra the corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra 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.
For each such cluster algebra we construct a large family of clusters and explicit mutations between them. Lie bialgebras, poisson lie groups and dressing transformations. Structure classical mechanics quantum mechanics space of states phase space m1 hilbert space h. Pdf vertex lie algebras, vertex poisson algebras and. The dual equation of a finitedimensional lie algebra equation. Poisson lie groups and lie bialgebras has been proved by drinfeld 6. Examples of poisson brackets weinstein symposium ihp, july 18 20, 20 jianghua lu. 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. Complete commutative subalgebras in polynomial poisson algebras. We show that a poisson structure can be induced on the affine moduli space of semisimple representations of an associative algebra from a suitable lie algebra structure on the zeroth hochschild homology of the algebra. We are also happy to publish dates and access data of other poisson.
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. Lie poisson space b, where b is the threedimensional book algebra so. The wellrecognized contributors to this text cover a broad. Noncoboundary poissonlie structures on the book group. His book, lie algebras, is a classic handbook both for researchers and students. If one has a lie bialgebra g, the hopf algebra sg is commutative and cocommutative, and has both a poisson bracket. In geometry, a poisson structure on a smooth manifold is a lie bracket. They are also called poissonhopf algebras, and are the lie algebra of a poissonlie group. To quantise functions on a group as a poisson algebra, quantise an appropriate dual.
Lie algebra of classical observables under poisson bracket. Poisson actions of g then correspond to hamiltonian actions of go together with their momentum maps. Though it presupposes knowledge of linear algebra, it is not overly theoretical and can be readily used for selfstudy. 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. Lie algebra, graded respectively, where the grading is shifted by, such that a graded version of the leibniz rule holds. This poisson structure on g will be of central importance from now on. The question of reconstructing the deformation a h from a poisson algebra is 2. Poisson geometry 225 this theorem unifies the two examples of complete poisson maps which were classified in section 6. Our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations. Poisson structures mathematical association of america. Liepoisson theory for direct limit lie algebras sciencedirect. Xiaomeng xu stokes phenomenon and poisson lie groups.
Poisson lie group admits a leftcovariant noncommutative di. 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. In particular this applies to necklace lie algebra for path algebras of doubled quivers and preprojective algebras. The algebra of a poissonlie group is a lie bialgebra. R poisson algebra are particular cases of poisson lie book groups. Check out the new look and enjoy easier access to your favorite features. Now we can put the above two stories on an equal footing. In mathematics, a poisson algebra is an associative algebra together with a lie bracket that also satisfies leibnizs law. Poisson lie pl groups are poisson structures on a lie group g for which the group multiplication is a poisson map. Keywords a generalized weyl poisson algebra a poisson algebra the poisson centre a poisson prime ideal the poisson simplicity. Here, is the lie algebra of the lie group and the linear mapping is defined to be.
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. Poisson lie algebra is the lie algebra of the quantomorphism group sch78, rs81. Their classification is fully performed by relating these pl groups to the corresponding lie bialgebra structures on the corresponding book lie algebra. We show that each triangular poisson lie group can be decomposed into poisson submanifolds each of which is a quotient of a symplectic manifold. We show that the two different qdeformed versions of the sl2. Poisson algebras appear naturally in hamiltonian mechanics, and are also central in the study of quantum groups. Alie bialgebrais a lie algebra g together with a compatiblelie algebra structure on g. The most classical example of a poisson algebra already considered by poisson himself is the algebra of smooth functions on r2nendowed with usual. A lie group with a poisson structure which is compatible with the group multiplication, i. 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. In mathematics, a poisson algebra is an associative algebra together with a lie bracket that. Triangular poisson structures on lie groups and symplectic reduction timothy j. 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. 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.
The starting point of liepoisson theory is the observation that the symplectic leaves of g. Needless to say, the poisson bracket preservation property has the same proof as 2. Triangular poisson structures on lie groups and symplectic. 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. Apoisson spaceis a space x with a poisson algebra structure on its algebra of functions.
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. The full set of poisson lie pl structures on the 3d book group are presented. Here we return to our motivating discussion of poisson manifolds, especially reasons 2 and 3 of. Twisting poisson algebras, copoisson algebras and quan tization. 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. Manifolds with a poisson algebra structure are known as poisson manifolds, of which the symplectic manifolds and the. The generalized weyl poisson algebras and their poisson. Examples of poisson brackets department of mathematics. One of the worlds foremost geometers, alan weinstein has made deep contributions to symplectic and differential geometry, lie theory, mechanics, and related fields. The marsdenweinsteinmeyer symplectic reduction technique is.
1053 1427 708 802 1022 147 1307 1437 1367 771 1385 641 144 594 1167 831 573 57 999 1335 1386 720 1166 1183 976 1013 1142 1186 580 1262 433 151 1344 955 1162