Poisson structures on moduli spaces of representations. Examples of poisson brackets department of mathematics. 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. A lecture on poissonnijenhuis structures springer for. Ane poisson structures on lie groups also serve as the target poisson structures for a not necessarily equivariant momentum mapping. It is proved that there is no nontrivial poisson hopf structure on the universal enveloping algebra of a nonabelian lie algebra. Onishchik, lie transformation groups encyclopaedia of mathematical sciences, volume 20, springer 1993. Let be an associative commutative algebra over a commutative ring cf. Free algebras, universal enveloping algebras, p th powers, uniqueness of restricted structures, existence of restricted structures, schemes, differential geometry of schemes, generalised witt algebra, filtrations, witt algebras are generalised witt algebra, differentials on a scheme, lie algebras of cartan type, root. For a detailed analysis of the structure and properties of poisson manifolds see also e. Said in another manner, it is a lie algebra structure on the vector space of smooth functions on such that. In both cases there 2000 mathematics subject classi.
Manifolds with a poisson algebra structure are known as poisson manifolds, of which the symplectic manifolds and the. Multiplicative and ane poisson structures on lie groups. The lie bracket v2 g gis regarded as an element of g. A poisson algebra structure on is defined by an bilinear skewsymmetric mapping such that.
In many cases, one considers the poisson structure on the lie algebra g itself, rather than on g. Let r be a commutative ring, and let a be a poisson algebra over r. This gives rise to suitable algebraic notions of poisson homology and cohomology for an arbitrary poisson algebra. It is a linear poisson structure and every linear poisson structure on a. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. We construct an r,a lie algebra structure, in the sense of rinehart, on the amodule of k\ahler differentials of a depending naturally on a and the poisson bracket. For a lie algebra the underlying dual vector space canonically inherits the structure of a poisson manifold whose poisson lie bracket. M of smooth functions under the poisson bracket to the lie algebra of smooth vector. Triangular poisson structures on lie groups and symplectic. The main difference between white noise analysis and the malliavin calculus is that the. The dual equation of a finitedimensional lie algebra equation. Prove that o2 and n2 are abelian 1dimensional lie algebras, hence they are isomorphic to kwith zero bracket. Symmetry free fulltext a stochastic poisson structure. A lie group with a poisson structure which is compatible with the group multiplication, i.
Similarly, one may look for deformations of a zinbiel algebra into dendriform algebras, and wonder. The dual \\mathfrakg\ of a finitedimensional lie algebra \\mathfrakg\ admits a natural poisson structure, called its lie poisson structure. 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 simone gutt, section 2. These are objects whose algebra of functions can be deformed into a hopf algebra. This lie algebra is unitarily equivalent to the usual spin physics angularmomentum component operators for spin1 particles in quantum mechanics. In mathematics, a poisson algebra is an associative algebra together with a lie bracket that also satisfies leibnizs law. Lie s book 11 contains a comprehensive theory of poisson manifolds under the name of function groups, including, among many results, a proof of the contravariant form of the jacobi identity, a proof of the duality between lie algebra structures and linear poisson. Some cohomology classes associated with an ideal in a lie algebra, a poisson structure on the basic functions algebra of contact structure, its poisson cohomology and geometric prequantization are considered from the algebraic point of view. A poisson bracket or a poisson structure on a manifold p is a bilinear operation, on fpc. Noncoboundary poissonlie structures on the book group article pdf available in journal of physics a mathematical and theoretical 4510 december 2011 with 67 reads how we measure reads. Drinfeld lie bialgebra structures on simple lie algebra. Lectures on poisson geometry department of mathematics. In particular this applies to necklace lie algebra for path algebras of doubled quivers and preprojective algebras.
Lie algebras have analogous properties, and the theory of lie algebroids uni. A poisson manifold endowed with a structure of a lie group is a poisson lie group if the multiplication map m. In geometry, a poisson structure on a smooth manifold is a lie bracket. 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. Multiplicative and affine poisson structures on lie groups hku.
The poisson structure on the group induces on the lie algebra an. Hence, we obtain a canonical quantization of the lie poisson structure, given essentially by the universal enveloping algebra. In this book, when we say that something is smooth without. Poisson brackets deform to moyal brackets upon quantization, that is, they generalize to a different lie algebra, the moyal algebra, or, equivalently in hilbert space, quantum commutators. Poisson manifolds poisson manifold was coined by lichnerowicz. We outline the role of derived brackets in the theory of poisson structures with background. The poisson bracket then makes ginto a poisson manifold. Note that poisson structures are a special case of the local lie algebra structures stud ied by kirillov 16. Modular lie algebras pdf 74p this note covers the following topics.
Pdf noncoboundary poissonlie structures on the book group. Poisson structures and their normal forms jeanpaul. An algebra, usually over the field of real or complex numbers, equipped with a bilinear mapping satisfying the properties of the usual poisson brackets of functions. Poisson algebra the corresponding infinitesimal object is a lie bialgebra see quantum groups, called the tangent lie bialgebra of. The construction of weinstein of a bialgebra structure on a compact semisimple lie algebra shows.
A lie group equipped with a multiplicative poisson structure is called a. The space of smooth vector fields on a differentiable manifold m forms a lie algebra, where the lie bracket is defined to be the. Here we return to our motivating discussion of poisson manifolds, especially reasons 2 and 3 of. G on g induces a lie algebra structure on t e g or equivalentlly a lie coalgebra structure on t eg by. Pdf poisson structures due to lie algebra representations krishna rama academia. It is a linear poisson structure and every linear poisson structure on a finitedimensional. The poisson algebra structure restricted to polynomial functions is described at the beginning of subsection 2. This poisson structure on g will be of central importance from now on. We the classify loworder extensions, thus showing that there are only a small number of independent. Prove that the lie algebra from example 2 is isomorphic to o3 by comparing the structure constants.
The question of whether every poisson structure admits a deformation quantization was. An important class of infinitedimensional real lie algebras arises in differential topology. Poisson structure an overview sciencedirect topics. Free lie algebra books download ebooks online textbooks. 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. Here, is the lie algebra of the lie group and the linear mapping is defined to be. We note that a poisson cgl extension is not a nilpotent lie algebra itself under the poisson bracket poisson algebras with such a property are very rare in the theory of poisson lie groups and poisson. 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 this sequel to 5, we study the behaviour of certain derived character maps of a dg lie algebra a with respect to a canonical derived. In what follows, lie conformal algebras will be playing the role of lie algebras in the usual nite dimensional theory. Linear poisson structures and lie algebras springerlink. Then, in the particular case of lie algebroids and differentiable manifolds, the poisson nijenhuis structures are related to the notion of a complementary 2form, that has been introduced and studied by the author in 20, and several examples of complementary forms and poisson nijenhuis manifolds are. Vice versa, if gis a lie algebra then its universal enveloping algebra ug is a poisson algebra by means of the same computation performed above. The general properties of such derived poisson structures have been studied in 7. Noncoboundary poissonlie structures on the book group article pdf available in journal of physics a mathematical and theoretical 4510 december 2011 with 67. Alie bialgebrais a lie algebra g together with a compatiblelie algebra structure on g. Stokes phenomenon and poisson lie groups mathematics. Poisson algebras appear naturally in hamiltonian mechanics, and are also central in the study of quantum groups. A poisson structure on a manifold p is a lie algebra operation, on. In each one of these fetchcontexts, it turns out that the poisson structure is not a theoretical artifact, but a key element which. 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.
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