# Poisson algebra

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, for all . The element is called the Poisson bracket of and . The main example is that of the algebra of smooth functions on a Poisson manifold [a5] (cf. also Symplectic structure).

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].