symmetric monoidal (∞,1)-category of spectra
A Poisson algebra is
The definition makes sense, but is not standardly used, also for the more general case when the product 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).
See there for more details.
For a Poisson manifold or a symplectic manifold, the algebra of smooth functions is naturally a Poisson algebra, thus may be regarded as an object in . For classical mechanical systems of this form, we say that the manifold is the phase space of the system.
Generally, therefore, for a Poisson algebra, we may regard it as a formal dual to some generalized Phase space.
If the Poisson algebra comes from a Poisson manifold , then this Lie-Rinehart pair is the Chevalley-Eilenberg algebra of the given Poisson Lie algebroid over . We can therefore identify classical mechanical systems over a phase space manifold also with Poisson Lie algebroids.
In terms of this, the Poisson bracket is given by
Write for the Lie algebra underlying the Poisson algebra.
This fits into a central extension of Lie algebras
This is also called the Kostant-Souriau central extension (see Kostant 1970).
A Poisson n-algebra is a Poisson algebra in chain complexes with a bracket of degree .
A Jordan-Lie-Banach algebra is a non-associative (quantum) Poisson algebra.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
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)