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. Let be an associative commutative algebra over a commutative ring (cf. Commutative algebra; Commutative ring; Associative rings and algebras). A Poisson algebra structure on is defined by an -bilinear skew-symmetric mapping such that
i) is a Lie algebra over ;
ii) the Leibniz rule is satisfied, namely,
On a Poisson algebra, one can define [a12] a skew-symmetric -bilinear mapping, , which generalizes the Poisson bivector on Poisson manifolds, mapping a pair of Kähler (or formal) differentials on to the algebra itself. There exists a unique -bilinear bracket, on the -module of Kähler differentials satisfying and lending it the structure of a Lie–Rinehart algebra, , for all . (Here, is the adjoint of , mapping the Kähler differentials into the derivations of ; cf. Adjoint operator.) The Poisson cohomology (cf. Cohomology) of is then defined and, when is projective as an -module, is equal to the cohomology of the complex of alternating -linear mappings on with values in , with the differential [a1] defined by the Lie–Rinehart algebra structure. In the case of the algebra of functions on a differentiable manifold, the Poisson cohomology coincides with the cohomology of the complex of multivectors, with differential , where is the Poisson bivector and is the Schouten bracket.
In a canonical ring [a4], the Poisson bracket is defined by a given mapping . Dirac structures [a13] on complexes over Lie algebras are a generalization of the Poisson algebras, adapted to the theory of infinite-dimensional Hamiltonian systems, where the ring of functions is replaced by the vector space of functionals.
In the category of -graded algebras, there are even and odd Poisson algebras, called graded Poisson algebras and Gerstenhaber algebras, respectively. Let be an associative, graded commutative algebra. A graded Poisson (respectively, Gerstenhaber) algebra structure on is a graded Lie algebra structure (cf. Lie algebra, graded) (respectively, where the grading is shifted by ), such that a graded version of the Leibniz rule holds: for each , is a derivation of degree (respectively, ) of the graded commutative algebra . Examples of Gerstenhaber algebras are: the Hochschild cohomology of an associative algebra [a2], in particular, the Schouten algebra of multivectors on a smooth manifold [a3], the exterior algebra of a Lie algebra, the algebra of differential forms on a Poisson manifold [a9], the space of sections of the exterior algebra of a Lie algebroid, the algebra of functions on an odd Poisson supermanifold of type [a7]. Batalin–Vil'koviskii algebras, also called BV-algebras, are exact Gerstenhaber algebras, i.e., their Lie bracket is a coboundary in the graded Hochschild cohomology of the algebra. Such structures arise on the BRST cohomology of topological field theories [a14].
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|[a11]||Y. Kosmann-Schwarzbach, F. Magri, "Poisson–Nijenhuis structures" Ann. Inst. H. Poincaré, Phys. Th. , 53 (1990) pp. 35–81|
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|[a13]||I. Dorfman, "Dirac structures and integrability of nonlinear evolution equations" , Wiley (1993)|
|[a14]||B.H. Lian, G.J. Zuckerman, "New perspectives on the BRST-algebraic structure of string theory" Comm. Math. Phys. , 154 (1993) pp. 613–646|
|[a15]||Y. Kosmann-Schwarzbach, "From Poisson to Gerstenhaber algebras" Ann. Inst. Fourier , 46 : 5 (1996) pp. 1243–1274|
|[a16]||M. Flato, M. Gerstenhaber, A.A. Voronov, "Cohomology and deformation of Leibniz pairs" Letters Math. Phys. , 34 (1995) pp. 77–90|
Poisson algebra. Y. Kosmann-Schwarzbach (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Poisson_algebra&oldid=17921