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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
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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" />;
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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  $  A $
 +
be an associative commutative algebra over a commutative ring  $  R $(
 +
cf. [[Commutative algebra|Commutative algebra]]; [[Commutative ring|Commutative ring]]; [[Associative rings and algebras|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|Lie algebra]] over $  R $;
  
 
ii) the Leibniz rule is satisfied, namely,
 
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>
+
$$
 +
\{ a,bc \} = \{ a,b \} c + b \{ a,c \}
 +
$$
  
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]]).
+
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 [[#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.
+
On a Poisson algebra, one can define [[#References|[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|Adjoint operator]].) The Poisson cohomology (cf. [[Cohomology|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 [[#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 $  d _ {P} = [ P, \cdot ] $,  
 +
where $  P $
 +
is the Poisson bivector and $  [ \cdot, \cdot ] $
 +
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 a canonical ring [[#References|[a4]]], the Poisson bracket is defined by a given mapping $  P  ^  \sharp  $.  
 +
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]]].
+
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|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 [[#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 $  ( n \mid  n ) $[[#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====
 
====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>
+
<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 {{ZBL|0356.17007}}</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>

Latest revision as of 19:18, 17 March 2023


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
How to Cite This Entry:
Poisson algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_algebra&oldid=17921
This article was adapted from an original article by Y. Kosmann-Schwarzbach (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article