# nLab Poisson algebra

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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).

Isbell 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{Gelfand-Kolmogorov}}{\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}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\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}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$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}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\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}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\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}$
• Yvette Kosmann-Schwarzbach, Poisson algebra, article in Encyclopedia of mathematics, (pdf)

• 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)