# 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 $A$ be an associative commutative algebra over a commutative ring $R$( cf. Commutative algebra; Commutative ring; Associative rings and algebras). A Poisson algebra structure on $A$ is defined by an $R$- bilinear skew-symmetric mapping ${\{ \cdot, \cdot \} } : {A \times A } \rightarrow A$ such that

i) $( A, \{ \cdot, \cdot \} )$ is a Lie algebra over $R$;

ii) the Leibniz rule is satisfied, namely,

$$\{ a,bc \} = \{ a,b \} c + b \{ a,c \}$$

for all $a, b, c \in A$. The element $\{ a,b \}$ is called the Poisson bracket of $a$ and $b$. 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 $A$- bilinear mapping, $P$, which generalizes the Poisson bivector on Poisson manifolds, mapping a pair of Kähler (or formal) differentials on $A$ to the algebra $A$ itself. There exists a unique $R$- bilinear bracket, $[ \cdot, \cdot ] _ {p}$ on the $A$- module $\Omega ^ {1} ( A )$ of Kähler differentials satisfying $[ da,db ] _ {P} = d \{ a,b \}$ and lending it the structure of a Lie–Rinehart algebra, $[ da,fdb ] _ {P} = f [ da,db ] _ {P} + P ^ \sharp ( da ) ( f ) db$, for all $a, b, f \in A$. (Here, $P ^ \sharp$ is the adjoint of $P$, mapping the Kähler differentials into the derivations of $A$; cf. Adjoint operator.) The Poisson cohomology (cf. Cohomology) of $A$ is then defined and, when $\Omega ^ {1} ( A )$ is projective as an $A$- module, is equal to the cohomology of the complex of alternating $A$- linear mappings on $\Omega ^ {1} ( A )$ with values in $A$, 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 $d _ {P} = [ P, \cdot ]$, where $P$ is the Poisson bivector and $[ \cdot, \cdot ]$ is the Schouten bracket.

In a canonical ring [a4], the Poisson bracket is defined by a given mapping $P ^ \sharp$. 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 $\mathbf Z$- graded algebras, there are even and odd Poisson algebras, called graded Poisson algebras and Gerstenhaber algebras, respectively. Let $A = \oplus A ^ {i}$ be an associative, graded commutative algebra. A graded Poisson (respectively, Gerstenhaber) algebra structure on $A$ is a graded Lie algebra structure (cf. Lie algebra, graded) $\{ \cdot, \cdot \}$( respectively, where the grading is shifted by $1$), such that a graded version of the Leibniz rule holds: for each $a \in A ^ {i}$, $\{ a, \cdot \}$ is a derivation of degree $i$( respectively, $i + 1$) of the graded commutative algebra $A = \oplus A ^ {i}$. 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 $( n \mid n )$[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].

#### References

 [a1] R.S. Palais, "The cohomology of Lie rings" , Proc. Symp. Pure Math. , 3 , Amer. Math. Soc. (1961) pp. 130–137 [a2] M. Gerstenhaber, "The cohomology structure of an associative ring" Ann. of Math. , 78 (1963) pp. 267–288 [a3] W.M. Tulczyjew, "The graded Lie algebra of multivector fields and the generalized Lie derivative of forms" Bull. Acad. Pol. Sci., Sér. Sci. Math. Astr. Phys. , 22 (1974) pp. 937–942 [a4] A. M. Vinogradov, I.S. Krasil'shchik, "What is the Hamiltonian formalism?" Russian Math. Surveys , 30 : 1 (1975) pp. 177–202 (In Russian) [a5] A. Lichnerowicz, "Les variétés de Poisson et leurs algèbres de Lie associées" J. Diff. Geom. , 12 (1977) pp. 253–300 [a6] J. Braconnier, "Algèbres de Poisson" C.R. Acad. Sci. Paris , A284 (1977) pp. 1345–1348 [a7] B. Kostant, "Graded manifolds, graded Lie theory and prequantization" K. Bleuler (ed.) A. Reetz (ed.) , Differential Geometric Methods in Mathematical Physics (Bonn, 1975) , Lecture Notes in Mathematics , 570 , Springer (1977) pp. 177–306 [a8] I.M. Gelfand, I.Ya. Dorfman, "Hamiltonian operators and algebraic structures related to them" Funct. Anal. Appl. , 13 (1979) pp. 248–262 (In Russian) [a9] J.-L. Koszul, "Crochet de Schouten–Nijenhuis et cohomologie" Astérisque, hors série, Soc. Math. France (1985) pp. 257–271 [a10] K.H. Bhaskara, K. Viswanath, "Calculus on Poisson manifolds" Bull. London Math. Soc. , 20 (1988) pp. 68–72 [a11] Y. Kosmann-Schwarzbach, F. Magri, "Poisson–Nijenhuis structures" Ann. Inst. H. Poincaré, Phys. Th. , 53 (1990) pp. 35–81 [a12] J. Huebschmann, "Poisson cohomology and quantization" J. Reine Angew. Math. , 408 (1990) pp. 57–113 [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
How to Cite This Entry:
Poisson algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_algebra&oldid=51639
This article was adapted from an original article by Y. Kosmann-Schwarzbach (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article