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 Zbl 0356.17007
[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