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Poisson algebra

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Definition

Definition

A Poisson algebra is

  • a module AA over some field or other commutative ring kk,

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

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

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

Definition

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

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

See there for more details.

Example

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

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

Remark

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

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

If the Poisson algebra comes from a Poisson manifold XX, then this Lie-Rinehart pair is the Chevalley-Eilenberg algebra of the given Poisson Lie algebroid over XX. 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,ω)(X, \omega) canonically is a Poisson manifold (X;{,})(X; \{-,-\}) by defining the Poisson bracket as follows.

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

d dRf=ι v fω. d_{dR} f = \iota_{v_f} \omega \,.

In terms of this, the Poisson bracket is given by

{f,g}:=ι v gι v fω. \{f,g\} := \iota_{v_g} \iota_{v_f} \omega \,.

Properties

For a symplectic manifold

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

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

Proposition

This fits into a central extension of Lie algebras

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

where Ham(X)Γ(TX)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).

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

References

Revised on July 12, 2012 17:04:44 by Urs Schreiber (89.204.139.213)