Springer Science & Business Media, Aug 27, 2012 - Mathematics - 464 pages
Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. 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). 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.
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Ad-invariant afﬁne algebra g arbitrary bialgebra bivector canonical Casimir function coefﬁcients commutative associative algebra consider constant coordinate chart deﬁned deﬁnition denote differential graded Lie dimension elements equation Example exists F and G ﬁeld ﬁrst following proposition formal deformation formal power series formula functions F g G G given graded Lie algebra Hamiltonian vector fields implies involution isomorphism Jacobi identity Lemma Let g Lie bracket Lie derivative Lie group Lie—Poisson structure linear functions linear map linear Poisson structure modular vector morphism neighborhood open subset order deformation Poisson algebra Poisson bracket Poisson cohomology Poisson manifold Poisson map Poisson matrix Poisson struc Poisson structure Poisson–Dirac Poisson–Lie group Proof quadratic Poisson structure R-matrix R-module rank respect satisﬁes Schouten bracket Section skew-symmetric biderivation structure on g symplectic leaf symplectic manifold tangent theorem vanishes vector space volume form weight homogeneous zero