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\begin{document}

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\section*{polynomial Poisson algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{noncommutative_geometry}{}\paragraph*{{Noncommutative geometry}}\label{noncommutative_geometry}

[[!include noncommutative geometry - contents]]

\hypertarget{formal_geometry}{}\paragraph*{{Formal geometry}}\label{formal_geometry}

[[!include formal geometry -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{formal_deformation_quantization_via_universal_enveloping_algebra}{Formal deformation quantization via universal enveloping algebra}\dotfill \pageref*{formal_deformation_quantization_via_universal_enveloping_algebra} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below). Remarkably, this statement generalizes to more general polynomial Poisson algebras (def. \ref{PolynomialPoissonAlgebra} below) for a suitable generalized concept of universal enveloping algebra (def. \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below): it is \emph{always} true up to third order in $\hbar$, and sometimes to higher order (\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, theorem 3.2}, prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3} below). In particular it also holds true for restrictions of [[Poisson bracket Lie algebras]] to their [[Heisenberg Lie algebras]] (example \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{PolynomialPoissonAlgebra}\hypertarget{PolynomialPoissonAlgebra}{}
\textbf{([[polynomial]] [[Poisson algebra]])}

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

\begin{displaymath}
Sym(V) \coloneqq T(V)/(x \otimes y - y \otimes x \vert x,y \in V)
\end{displaymath}
on some [[vector space]] $V$. Here

\begin{displaymath}
T(V) \coloneqq \underset{n \in \mathbb{N}}{\oplus} V^{\otimes^n}
\end{displaymath}
denotes the [[tensor algebra]] of $V$. We write

\begin{displaymath}
\mu \;\colon\; T(V) \longrightarrow Sym(V)
\end{displaymath}
for the canonical [[projection]] map (which is an algebra [[homomorphism]]) and

\begin{displaymath}
\sigma \;\colon\; Sym(V) \longrightarrow T(V)
\end{displaymath}
for its [[linear map|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

\begin{displaymath}
\{-,-\} \;\colon\; V \otimes V \longrightarrow Sym(V)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}\hypertarget{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}{}
\textbf{(universal enveloping algebra of polynomial Poisson algebra)}

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

\begin{displaymath}
\mathcal{U}(V,\{-,-\})
  \;\coloneqq\;
  T(V)/( x \otimes y - y \otimes x - \hbar \{x,y\} \vert x,y \in V )
  \,.
\end{displaymath}
This comes with the quotient projection linear map which we denote by

\begin{displaymath}
\rho \;\colon\; T(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\hbar\{-,-\})
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, def. 3.1})

The combined linear projection maps from def. \ref{PolynomialPoissonAlgebra} and def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra} we denote by

\begin{displaymath}
\tau \coloneqq \rho \circ (\sigma/[ [ \hbar ] ]) \;\colon\; Sym(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\{-,-\})
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{PolynomialLiePoissonStructure}\hypertarget{PolynomialLiePoissonStructure}{}
\textbf{([[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]] $\mathbb{R}[x^1, \cdots, x^n ] \ookrightarrow C^\infty(\mathbb{R}^n)$ is a polynomial Poisson algebra according to def. \ref{PolynomialPoissonAlgebra}.

In particular if $(\mathfrak{g}, [-,-])$ is a [[Lie algebra]] and $(\mathfrak{g}^\ast, \{-,-\})$ the corresponding [[Lie-Poisson structure|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]]:

\begin{displaymath}
\{x,y\} = [x,y]
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{UniversalEnvelopingAlgebraOfLieAlgebra}\hypertarget{UniversalEnvelopingAlgebraOfLieAlgebra}{}
\textbf{([[universal enveloping algebra]] of [[Lie algebra]])}

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

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{formal_deformation_quantization_via_universal_enveloping_algebra}{}\subsubsection*{{Formal deformation quantization via universal enveloping algebra}}\label{formal_deformation_quantization_via_universal_enveloping_algebra}

\begin{prop}
\label{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}\hypertarget{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}{}
\textbf{(universal enveloping algebra provides [[deformation quantization]] at least up to order 3)}

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

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

\end{prop}
(\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, theorem 3.2 with section 2})

\begin{example}
\label{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras}\hypertarget{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras}{}
\textbf{([[formal deformation quantization]] of [[Lie-Poisson structures]] by universal enveloping algebras)}

In the following cases the map $\tau$ in prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3} is injective to arbitrary order, hence in these cases the universal enveloping algebra provides a genuine [[deformation quantization]]:

\begin{enumerate}%
\item the case that the Poisson bracket is [[linear Poisson structure|linear]] in that restricts as

\begin{displaymath}
\{-,-\} \;\colon\; V \otimes V \longrightarrow V \hookrightarrow Sym(V)
  \,.
\end{displaymath}
This is the case of the [[Lie-Poisson structure]] from example \ref{PolynomialLiePoissonStructure} and the universal enveloping algebra that provides it deformation quantization is the standard one (example \ref{UniversalEnvelopingAlgebraOfLieAlgebra}).


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

\begin{displaymath}
\{-,-\} \;\colon\; V \otimes V \longrightarrow \mathbb{R} \oplus V \hookrightarrow Sym(V)
  \,.
\end{displaymath}
This includes notably the Poisson structures induced by [[symplectic vector spaces]], in which case the restriction

\begin{displaymath}
\{-,-\} \;\colon\; (\mathbb{R} \oplus V) \otimes (\mathbb{R} \oplus V) \longrightarrow (\mathbb{R} \oplus V)
\end{displaymath}
is the [[Lie bracket]] of the associated [[Heisenberg Lie algebra]].



\end{enumerate}
\end{example}
This is (\hyperlink{PenkavaVanhaecke00}{Penkava-Vanhaecke 00, p. 26}) The first statement in itself is a classical fact (reviewed e.g. in \hyperlink{Gutt11}{Gutt 11}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Michael Penkava]], [[Pol Vanhaecke]], \emph{Deformation quantization of Polynomial Poisson algebras}, Journal of Algebra 227, 365-393 2000 (\href{https://arxiv.org/abs/math/9804022}{arXiv:math/9804022})

\end{itemize}
[[!redirects polynomial Poisson algebras]]



\end{document}