nLab polynomial Poisson algebra

Idea

It is a classical fact that the universal enveloping algebra of a Lie algebra provides a formal deformation quantization of the corresponding Lie-Poisson structure (example below). Remarkably, this statement generalizes to more general polynomial Poisson algebras (def. below) for a suitable generalized concept of universal enveloping algebra (def. below): it is always true up to third order in $\hbar$, and sometimes to higher order (Penkava-Vanhaecke 00, theorem 3.2, prop. below). In particular it also holds true for restrictions of Poisson bracket Lie algebras to their Heisenberg Lie algebras (example below).

Definition

Definition

(polynomial Poisson algebra)

A Poisson algebra $((A,\cdot), \{-,-\})$ is called a polynomial Poisson algebra if the underlying commutative algebra $(A,\cdot)$ is a polynomial algebra, hence a symmetric algebra

$$Sym(V) \coloneqq T(V)/(x \otimes y - y \otimes x \vert x,y \in V)$$

on some vector space $V$. Here

$$T(V) \coloneqq \underset{n \in \mathbb{N}}{\oplus} V^{\otimes^n}$$

denotes the tensor algebra of $V$. We write

$$\mu \;\colon\; T(V) \longrightarrow Sym(V)$$

for the canonical projection map (which is an algebra homomorphism) and

$$\sigma \;\colon\; Sym(V) \longrightarrow T(V)$$

for its linear inverse (symmetrization, which is not in general an algebra homomorphism).

Notice that by its bi-derivation property the Poisson bracket on a polynomial Poisson algebra is fixed by its restriction to linear elements

$$\{-,-\} \;\colon\; V \otimes V \longrightarrow Sym(V) \,.$$

Definition

(universal enveloping algebra of polynomial Poisson algebra)

Given a polynomial Poisson algebra $(Sym(V), \{-,-\})$ (def. ), say that its universal enveloping algebra $\mathcal{U}(V,\{-,-\})$ is the associative algebra which is the quotient of the tensor algebra of $V$ with a formal variable $\hbar$ adjoined by the two-sided ideal which is generated by the the $\hbar$-Poisson bracket relation on linear elements:

$$\mathcal{U}(V,\{-,-\}) \;\coloneqq\; T(V)/( x \otimes y - y \otimes x - \hbar \{x,y\} \vert x,y \in V ) \,.$$

This comes with the quotient projection linear map which we denote by

$$\rho \;\colon\; T(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\hbar\{-,-\}) \,.$$

(Penkava-Vanhaecke 00, def. 3.1)

The combined linear projection maps from def. and def. we denote by

$$\tau \coloneqq \rho \circ (\sigma/[ [ \hbar ] ]) \;\colon\; Sym(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\{-,-\}) \,.$$

Examples

Example

(Lie-Poisson structure on affine algebraic variety)

Let $(C^\infty(\mathbb{R}^n), \pi)$ be a Poisson manifold whose underlying manifold is a Cartesian space $\mathbb{R}^n$. Then the restriction of its Poisson algebra $( C^\infty(\mathbb{R}^n, \cdot), \pi^{i j} \partial_i(-) \cdot \partial_j(-) )$ to the polynomial functions $\mathbb{R}[x^1, \cdots, x^n ] \hookrightarrow C^\infty(\mathbb{R}^n)$ is a polynomial Poisson algebra according to def. .

In particular if $(\mathfrak{g}, [-,-])$ is a Lie algebra and $(\mathfrak{g}^\ast, \{-,-\})$ the corresponding Lie-Poisson manifold, then the corresponding polynomial Poisson algebra is $(Sym(\mathfrak{g}), \{-,-\})$ where the restriction of the Poisson bracket to linear polynomial elements coincides with the Lie bracket:

$$\{x,y\} = [x,y] \,.$$

Example

(universal enveloping algebra of Lie algebra)

In the case of a polynomial Lie-Poisson structure $(Sym(\mathfrak{g}), [-,-])$ (example ) the universal enveloping algebra $\mathcal{U}(\mathfrak{g},[-,-])$ from def. (for $\hbar = 1$) coincides with the standard universal enveloping algebra of the Lie algebra $(\mathfrak{g}, [-,-])$.

Properties

Formal deformation quantization via universal enveloping algebra

Proposition

(universal enveloping algebra provides deformation quantization at least up to order 3)

Let $( Sym(V), \{-,-\} )$ be a polynomial Poisson algebra (def. ) such that the canonical linear map to its universal enveloping algebra (def. ) is injective up to order $n \in \mathbb{N}\cup \{\infty\}$

$$\tau/(\hbar^{n+1}) \;\colon\; Sym(V)[ [ \hbar ] ]/(\hbar^{n+1}) \hookrightarrow \mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1}) \,.$$

Then the restriction of the product on $\mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1})$ to $Sym(V)/(\hbar^{n+1})$ is a deformation quantization of $(Sym(V), \{-,-\})$ to order $n$ (hence a genuine deformation quantization in the case that $n = \infty$).

Moreover, this is always the case for $n = 3$, hence for every polynomial Poisson algebra its universal enveloping algebra always provides a deformation quantization of order $3$ in $\hbar$.

(Penkava-Vanhaecke 00, theorem 3.2 with section 2)

Example

(formal deformation quantization of Lie-Poisson structures by universal enveloping algebras)

In the following cases the map $\tau$ in prop. is injective to arbitrary order, hence in these cases the universal enveloping algebra provides a genuine deformation quantization:

1. the case that the Poisson bracket is linear in that restricts as

   $$\{-,-\} \;\colon\; V \otimes V \longrightarrow V \hookrightarrow Sym(V) \,.$$

   This is the case of the Lie-Poisson structure from example and the universal enveloping algebra that provides it deformation quantization is the standard one (example ).

2. more generally, the case that the Poisson bracket restricted to linear elements has linear and constant contribution in that it restricts as

   $$\{-,-\} \;\colon\; V \otimes V \longrightarrow \mathbb{R} \oplus V \hookrightarrow Sym(V) \,.$$

   This includes notably the Poisson structures induced by symplectic vector spaces, in which case the restriction

   $$\{-,-\} \;\colon\; (\mathbb{R} \oplus V) \otimes (\mathbb{R} \oplus V) \longrightarrow (\mathbb{R} \oplus V)$$

   is the Lie bracket of the associated Heisenberg Lie algebra.

This is (Penkava-Vanhaecke 00, p. 26) The first statement in itself is a classical fact (reviewed e.g. in Gutt 11).

References

• Michael Penkava, Pol Vanhaecke, Deformation quantization of Polynomial Poisson algebras, Journal of Algebra 227, 365-393 2000 (arXiv:math/9804022)