nLab Poisson algebra

Definition

Definition

A Poisson algebra is

• a module $A$ over some field or other commutative ring $k$,

• equipped with the structure ${\cdot}\colon A \otimes_k A \to A$ of a commutative associative algebra;

• and equipped with the bracket $[-,-]\colon A \otimes_k A \to A$ of a Lie algebra;

• such that for every $a \in A$ we have that $[a,-]\colon A \to A$ is a derivation of $(A,\cdot)$.

The definition makes sense, but is not standardly used, also for the more general case when the product $\cdot$ is not necessarily commutative (it is often however taken in the sense of commutative internally to a symmetric monoidal category, say of chain complexes, graded vector spaces or supervector spaces).

Poisson algebras form a category Poiss.

Definition

The opposite category of that of (commutative) real Poisson algebras can be identified with the category of classical mechanical systems

$$ClassMechSys := CPoiss^{op} \,.$$

See there for more details.

Example

For $(X, \{-,-\})$ a Poisson manifold or $(X, \omega)$ a symplectic manifold, the algebra of smooth functions $C^\infty(X, \mathbb{R})$ is naturally a Poisson algebra, thus may be regarded as an object in $ClassMechSys$. For classical mechanical systems of this form, we say that the manifold $X$ is the phase space of the system.

Generally, therefore, for $(A, \cdot,[-,-])$ a Poisson algebra, we may regard it as a formal dual to some generalized Phase space.

Remark

For $(A, \cdot, \{-,-\})$ a Poisson algebra, $A$ together with its module $\Omega^1(A)$ of Kähler differentials naturally form a Lie-Rinehart pair, with bracket given by

$$[d a, d b ] := d \{a,b\} \,.$$

If the Poisson algebra comes from a Poisson manifold $X$, then this Lie-Rinehart pair is the Chevalley-Eilenberg algebra of the given Poisson Lie algebroid over $X$. We can therefore identify classical mechanical systems over a phase space manifold also with Poisson Lie algebroids.

Examples

For a symplectic manifold

Definition

A symplectic manifold $(X, \omega)$ canonically is a Poisson manifold $(X; \{-,-\})$ by defining the Poisson bracket as follows.

By the symplectic structure, to every smooth function $f \in C^\infty(X)$ is associated the correspinding Hamiltonian vector field $v_f \in \Gamma(T X)$, defined, uniquely, by the equation

$$d_{dR} f = \iota_{v_f} \omega \,.$$

In terms of this, the Poisson bracket is given by

$$\{f,g\} := \iota_{v_g} \iota_{v_f} \omega \,.$$

Properties

For a symplectic manifold

Let $(X, \omega)$ be a symplectic manifold, and $(X, \{-,-\})$ the corresponding Poisson manifold as above.

Write $\mathcal{P} := (C^\infty(X), \{-,-\})$ for the Lie algebra underlying the Poisson algebra.

Proposition

This fits into a central extension of Lie algebras

$$\mathbb{R} \to \mathcal{P} \to Ham(X) \,,$$

where $Ham(X) \subset \Gamma(T X)$ is the sub-Lie algebra of vector fields on the Hamiltonian vector fields.

Proof

Observe that the Hamiltonian function associated to a Hamiltonian vector field is well-defined only up to addition of a constant function.

This is also called the Kostant-Souriau central extension (see Kostant 1970).

Related concepts

• A polynomial Poisson algebra is one whose underlying commutative algebra is a polynomial algebra.

• A Poisson manifold is a smooth manifold whose associatve algebra of smooth functions with values in the real line $\mathbb{R}$ that is equipped with the structure of a Poisson algebra (over $\mathbb{R}$).

• A Poisson n-algebra is a Poisson algebra in chain complexes with a bracket of degree $1-n$.

  • A Gerstenhaber algebra is a Poisson 2-algebra.

• A Coisson algebra is essentially a Poisson algebra internal to D-modules.

• A Jordan–Lie algebra is a non-associative (quantum) Poisson algebra.

• The quantomorphism group is the Lie group that integrates the Poisson Lie bracket. Over a symplectic vector space this contains notably the corresponding Heisenberg group.

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

References

• Yvette Kosmann-Schwarzbach, Poisson algebra, article in Encyclopedia of mathematics, (pdf)

• Klaas Landsman, Mathematical Topics Between Classical and Quantum Mechanics

• N. Chriss, Victor Ginzburg, Complex geometry and representation theory

• Peter Olver, Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995

• Bertram Kostant, Quantization and unitary representations. I. Prequantization, In Lectures in Modern Analysis and Applications, III, pages 87–208. Lecture Notes in Math., Vol. 170. Springer, Berlin (1970)

Discussion of Lie integration of the Poisson brackets on non-compact manifolds, by restriction to compactly supported functions:

• Augustin Banyaga, Paul Donato, Some remarks on the integration of the Poisson algebra, Journal of Geometry and Physics 19 4 (1996) 368-378 [doi:10.1016/0393-0440(95)00039-9]