View source for Poisson algebra

An [[Algebra|algebra]], usually over the field of real or complex numbers, equipped with a [[Bilinear mapping|bilinear mapping]] satisfying the properties of the usual [[Poisson brackets|Poisson brackets]] of functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101701.png" /> be an associative commutative algebra over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101702.png" /> (cf. [[Commutative algebra|Commutative algebra]]; [[Commutative ring|Commutative ring]]; [[Associative rings and algebras|Associative rings and algebras]]). A Poisson algebra structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101703.png" /> is defined by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101704.png" />-bilinear skew-symmetric mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101705.png" /> such that

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101706.png" /> is a [[Lie algebra|Lie algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101707.png" />;

ii) the Leibniz rule is satisfied, namely,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101708.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p1101709.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017010.png" /> is called the Poisson bracket of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017012.png" />. The main example is that of the algebra of smooth functions on a Poisson manifold [[#References|[a5]]] (cf. also [[Symplectic structure|Symplectic structure]]).

On a Poisson algebra, one can define [[#References|[a12]]] a skew-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017013.png" />-bilinear mapping, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017014.png" />, which generalizes the Poisson bivector on Poisson manifolds, mapping a pair of Kähler (or formal) differentials on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017015.png" /> to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017016.png" /> itself. There exists a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017017.png" />-bilinear bracket, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017018.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017019.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017020.png" /> of Kähler differentials satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017021.png" /> and lending it the structure of a Lie–Rinehart algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017022.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017023.png" />. (Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017024.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017025.png" />, mapping the Kähler differentials into the derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017026.png" />; cf. [[Adjoint operator|Adjoint operator]].) The Poisson cohomology (cf. [[Cohomology|Cohomology]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017027.png" /> is then defined and, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017028.png" /> is projective as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017029.png" />-module, is equal to the cohomology of the complex of alternating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017030.png" />-linear mappings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017031.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017032.png" />, with the differential [[#References|[a1]]] defined by the Lie–Rinehart algebra structure. In the case of the algebra of functions on a [[Differentiable manifold|differentiable manifold]], the Poisson cohomology coincides with the cohomology of the complex of multivectors, with differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017034.png" /> is the Poisson bivector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017035.png" /> is the Schouten bracket.

In a canonical ring [[#References|[a4]]], the Poisson bracket is defined by a given mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017036.png" />. Dirac structures [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017037.png" />-graded algebras, there are even and odd Poisson algebras, called graded Poisson algebras and Gerstenhaber algebras, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017038.png" /> be an associative, graded commutative algebra. A graded Poisson (respectively, Gerstenhaber) algebra structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017039.png" /> is a graded Lie algebra structure (cf. [[Lie algebra, graded|Lie algebra, graded]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017040.png" /> (respectively, where the grading is shifted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017041.png" />), such that a graded version of the Leibniz rule holds: for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017043.png" /> is a derivation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017044.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017045.png" />) of the graded commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017046.png" />. Examples of Gerstenhaber algebras are: the Hochschild cohomology of an associative algebra [[#References|[a2]]], in particular, the Schouten algebra of multivectors on a smooth manifold [[#References|[a3]]], the exterior algebra of a Lie algebra, the algebra of differential forms on a Poisson manifold [[#References|[a9]]], the space of sections of the exterior algebra of a Lie algebroid, the algebra of functions on an odd Poisson supermanifold of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110170/p11017047.png" /> [[#References|[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 [[#References|[a14]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.S. Palais,   "The cohomology of Lie rings" , ''Proc. Symp. Pure Math.'' , '''3''' , Amer. Math. Soc.  (1961)  pp. 130–137</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Gerstenhaber,   "The cohomology structure of an associative ring"  ''Ann. of Math.'' , '''78'''  (1963)  pp. 267–288</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. M. Vinogradov,   I.S. Krasil'shchik,   "What is the Hamiltonian formalism?"  ''Russian Math. Surveys'' , '''30''' :  1  (1975)  pp. 177–202  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Lichnerowicz,   "Les variétés de Poisson et leurs algèbres de Lie associées"  ''J. Diff. Geom.'' , '''12'''  (1977)  pp. 253–300</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Braconnier,   "Algèbres de Poisson"  ''C.R. Acad. Sci. Paris'' , '''A284'''  (1977)  pp. 1345–1348</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  I.M. Gelfand,   I.Ya. Dorfman,   "Hamiltonian operators and algebraic structures related to them"  ''Funct. Anal. Appl.'' , '''13'''  (1979)  pp. 248–262  (In Russian)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.-L. Koszul,   "Crochet de Schouten–Nijenhuis et cohomologie"  ''Astérisque, hors série, Soc. Math. France''  (1985)  pp. 257–271</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  K.H. Bhaskara,   K. Viswanath,   "Calculus on Poisson manifolds"  ''Bull. London Math. Soc.'' , '''20'''  (1988)  pp. 68–72</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  Y. Kosmann-Schwarzbach,   F. Magri,   "Poisson–Nijenhuis structures"  ''Ann. Inst. H. Poincaré, Phys. Th.'' , '''53'''  (1990)  pp. 35–81</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Huebschmann,   "Poisson cohomology and quantization"  ''J. Reine Angew. Math.'' , '''408'''  (1990)  pp. 57–113</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  I. Dorfman,   "Dirac structures and integrability of nonlinear evolution equations" , Wiley  (1993)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  B.H. Lian,   G.J. Zuckerman,   "New perspectives on the BRST-algebraic structure of string theory"  ''Comm. Math. Phys.'' , '''154'''  (1993)  pp. 613–646</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  Y. Kosmann-Schwarzbach,   "From Poisson to Gerstenhaber algebras"  ''Ann. Inst. Fourier'' , '''46''' :  5  (1996)  pp. 1243–1274</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  M. Flato,   M. Gerstenhaber,   A.A. Voronov,   "Cohomology and deformation of Leibniz pairs"  ''Letters Math. Phys.'' , '''34'''  (1995)  pp. 77–90</TD></TR></table>