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\section*{Lie-Poisson structure}

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

[[!include symplectic 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{abstractly}{Abstractly}\dotfill \pageref*{abstractly} \linebreak
\noindent\hyperlink{DefinitionInComponents}{In components}\dotfill \pageref*{DefinitionInComponents} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{deformation_quantization_by_universal_enveloping_algebra}{Deformation quantization by universal enveloping algebra}\dotfill \pageref*{deformation_quantization_by_universal_enveloping_algebra} \linebreak
\noindent\hyperlink{symplectic_groupoid}{Symplectic groupoid}\dotfill \pageref*{symplectic_groupoid} \linebreak
\noindent\hyperlink{symplectic_leaves}{Symplectic leaves}\dotfill \pageref*{symplectic_leaves} \linebreak
\noindent\hyperlink{PoissonLieAlgebroidCohomology}{Poisson-Lie algebroid cohomology}\dotfill \pageref*{PoissonLieAlgebroidCohomology} \linebreak
\noindent\hyperlink{poisson_lie_group_structure}{Poisson Lie group structure}\dotfill \pageref*{poisson_lie_group_structure} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $\mathfrak{g}$ a [[Lie algebra]] the underlying [[dual vector space]] $\mathfrak{g}^*$ canonically inherits the structure of a [[Poisson manifold]] whose Poisson [[Lie bracket]] reduces on linear functions $\mathfrak{g} \hookrightarrow C^\infty(\mathfrak{g}^*)$ to the original Lie bracket on $\mathfrak{g}$. This is the \textbf{Lie-Poisson structure} on $\mathfrak{g}^*$.

More generally, for $\mathfrak{a}$ a [[Lie algebroid]] the fiberwise dual $\mathfrak{a}^*$ inherits such a Poisson manifold structure.

[[Poisson manifold]] structures of this form are also called \emph{[[linear Poisson structures]]}.

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

\hypertarget{abstractly}{}\subsubsection*{{Abstractly}}\label{abstractly}

First notice that for $f \in C^\infty(\mathfrak{g}^\ast)$ as smooth function on the dual of a Lie algebra, then its [[de Rham differential]] 1-form at some $\alpha \in \mathfrak{g}^\ast$, being a linear map

\begin{displaymath}
\mathbf{d} f|_{\alpha}
  \colon
  T_\alpha \mathfrak{g}^\ast
  =
  \mathfrak{g}^\ast
  \longrightarrow
  \mathbb{R}
\end{displaymath}
is canonically identified with a Lie algebra element itself.

With this understood, then for $f,g \in C^\infty(\mathfrak{g}^*)$ two [[smooth functions]] on $\mathfrak{g}^*$ their Poisson [[Lie bracket]] in the Lie-Poisson structure is defined by

\begin{displaymath}
\{f,g\} \;\colon\; \theta \mapsto -\theta ([\mathbf{d} f, \mathbf{d} g])
  \,.
\end{displaymath}
Notice that for $v\in \mathfrak{g}$ regarded as a linear function $\langle -,v\rangle$ on $\mathfrak{g}^\ast$, then under the above identification we have $\mathbf{d} \langle -,v\rangle = v$. This means that on linear functions the Lie-Poisson bracket is simply the original Lie bracket:

\begin{displaymath}
\left\{
    \langle -, v_1\rangle,
    \langle -, v_2\rangle,
  \right\}
  = 
  \langle - ,[v_1,v_2]\rangle
  \,.
\end{displaymath}
This Lie-Poisson structure may be thought of as the unique smooth extension of this bracket on linear functions to all smooth functions on $\mathfrak{g}^\ast$.

\hypertarget{DefinitionInComponents}{}\subsubsection*{{In components}}\label{DefinitionInComponents}

Let $\{x^a\}$ be a [[basis]] for the [[vector space]] underlying the given [[Lie algebra]] $\mathfrak{g}$. Write $\{C^{a b}{}_c\}$ for the components of the [[Lie bracket]] $[-,-]$ in this basis (the structure constants), given by

\begin{displaymath}
[x^a,x^b] = \underset{c}{\sum} C^{a b}{}_c x^c
  \,.
\end{displaymath}
Write $\{\partial_a\}$ for the dual basis of the [[dual vector space]] $\mathfrak{g}^\ast$, so that the pairing $\mathfrak{g}^\ast \otimes\mathfrak{g} \to \mathbb{R}$ is given by

\begin{displaymath}
\partial_a x^b = \delta_a^b = 
  \left\{
    \itexarray{
      1 & if\; a=b
      \\
      0 & otherwise
    }
  \right.
\end{displaymath}
As the notation is meant to suggest, dually the $\{x^a\}$ may be regarded as basis for the linear functions on $\mathfrak{g}^\ast$ and the $\{\partial_a\}$ serve as a basis of [[vector fields]] on $\mathfrak{g}^\ast$.

With this identification understood, the [[multivector fields]] on $\mathfrak{g}^\ast$ are spanned by elements of the form

\begin{displaymath}
v^{a_1 \cdots a_q} \partial_{a_1}\wedge \cdots \wedge \partial_{a_q}
\end{displaymath}
(with the sum over indices understood) for $\{v^{a_1 \cdots a_q}\}$ smooth functions on $\mathfrak{g}^\ast$.

The [[Poisson tensor]] $\pi \in \wedge^2 \Gamma(T\mathfrak{g}^\ast)$ of the Lie-Poisson structure is given by

\begin{displaymath}
\pi = \tfrac{1}{2}\underset{a,b,c}{\sum} C^{a b}{}_c x^c \partial_a \wedge \partial_b
  \,.
\end{displaymath}
The [[Schouten bracket]] on multivector fields is given on linear basis elements by

\begin{displaymath}
\{x^a, x^b\}_{Sch} = 0
\end{displaymath}
\begin{displaymath}
\{\partial_a, x^b\}_{Sch} = \delta_a^b
\end{displaymath}
\begin{displaymath}
\{\partial_a, \partial_b\}_{Sch} = 0
\end{displaymath}
(the [[canonical commutation relations]]) and extended as a graded [[derivation]] in both arguments.

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

\hypertarget{deformation_quantization_by_universal_enveloping_algebra}{}\subsubsection*{{Deformation quantization by universal enveloping algebra}}\label{deformation_quantization_by_universal_enveloping_algebra}

See at \emph{[[deformation quantization]]} the section \emph{\href{deformation+quantization#RelationToUniversalEnvelopingAlgebras}{Relation to universal enveloping algebras}}.

\hypertarget{symplectic_groupoid}{}\subsubsection*{{Symplectic groupoid}}\label{symplectic_groupoid}

The [[symplectic groupoid]] [[Lie integration|integrating]] the Lie-Poisson structure on $\mathfrak{g}^*$ is the [[action groupoid]] $\mathfrak{g}^* //G$ of the [[coadjoint action]]. For more see at \emph{[[symplectic groupoid]]} in the section \emph{\href{symplectic+groupoid#OfLiePoissonStructure}{Examples -- Of Lie-Poisson stucture}}.

\hypertarget{symplectic_leaves}{}\subsubsection*{{Symplectic leaves}}\label{symplectic_leaves}

The [[symplectic leaves]] of the Lie-Poisson structure on $\mathfrak{g}^*$ are the [[coadjoint orbits]].

\hypertarget{PoissonLieAlgebroidCohomology}{}\subsubsection*{{Poisson-Lie algebroid cohomology}}\label{PoissonLieAlgebroidCohomology}

We consider the [[Poisson Lie algebroid]] $\mathfrak{P}(\mathfrak{g}^\ast)$ of a Lie-Poisson structure and the [[Lie algebroid cohomology]].

\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[Poisson Lie algebroid]]}, the graded algebra of [[multivector fields]] equipped with the [[differential]] given by the [[Schouten bracket]] with the [[Poisson bivector]]

\begin{displaymath}
d_{CE} = \{\pi, -\}_{Sch}
\end{displaymath}
is the [[Chevalley-Eilenberg algebra]] of this Lie algebroid:

\begin{displaymath}
CE(\mathfrak{P}(\mathfrak{g}^\ast))
  = 
  \left(
    \wedge^\bullet \Gamma(T\mathfrak{g}^\ast),
    d_{CE} = \{\pi, -\}_{Sch}
  \right)
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
As for every [[Poisson Lie algebroid]], the [[Poisson bivector]] $\pi \in CE(\mathfrak{P}(\mathfrak{g}^\ast))$ is a [[Lie algebroid cocycle]] of degree 2

\begin{displaymath}
d_{CE}\pi = \{\pi,\pi\}_{Sch} = 0
\end{displaymath}
(see also at \emph{[[symplectic Lie n-algebroid]]}).

In view of the fact that here $\pi$ is just another incarnation of the [[Lie bracket]], this condition here is an incarnation of the [[Jacobi identity]] on the Lie algebra $(\mathfrak{g},[-,-])$.

\end{remark}
But in the simple case of Lie-Poisson structure, this cocycle is in fact exact:

\begin{prop}
\label{PoissonTensorCoboundary}\hypertarget{PoissonTensorCoboundary}{}
For the Poisson-Lie structure on $\mathfrak{g}^\ast$ the [[Poisson tensor]] $\pi \in CE^2(\mathfrak{P}(\mathfrak{g}))$ has a [[coboundary]] and hence is trivial in [[Lie algebroid cohomology]].

\end{prop}
\begin{proof}
Consider the component-description from \hyperlink{DefinitionInComponents}{above}. We show that $x^a \partial_a$ is a coboundary.

First notice that

\begin{displaymath}
\{x^b, x^a \partial_a \}_{Sch} = -x^b
\end{displaymath}
and

\begin{displaymath}
\{\partial_b, x^a \partial_a \}_{Sch} = \partial_b
  \,.
\end{displaymath}
From this we get

\begin{displaymath}
\begin{aligned}
    d_{CE} (x^a \partial_a)
    &= 
   \left\{\pi,\;  x^a \partial_a\right\}_{Sch}
    \\
    & = 
    \left\{\frac{1}{2} C^{a b}{}_c x^c \partial_a \wedge \partial_b,\;    
    x^a \partial_a\right\}_{Sch}    
    \\
    & = (2-1) \frac{1}{2} C^{a b}{}_c x^c \partial_a \wedge \partial_b
    \\
    & = 
    \pi
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{poisson_lie_group_structure}{}\subsubsection*{{Poisson Lie group structure}}\label{poisson_lie_group_structure}

Under addition a Lie-Poisson manifold becomes a \emph{[[Poisson Lie group]]}, see there for more.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item a \emph{[[moment map]]} is often expresses as a Poisson homomorphism into a Lie-Poisson structure.

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

The notion of Lie-Poisson structures was originally found by [[Sophus Lie]] and then rediscovered by [[Felix Berezin]] and by [[Alexander Kirillov]], [[Bertram Kostant]] and [[Jean-Marie Souriau]].

General accounts include

\begin{itemize}%
\item Izu Vaisman, section 3.1 of \emph{Lectures on the Geometry of Poisson Manifolds}, Birkh\"a{}user 1994


\item [[Camille Laurent-Gengoux]], \emph{Linear Poisson Structures and Lie Algebras}, chapter 7 pp 179-203 of Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke (eds.) \emph{Poisson Structures}, Grundlehren der mathematischen Wissenschaften book series (GL, volume 347) (\href{https://link.springer.com/chapter/10.1007/978-3-642-31090-4_7}{web})



\end{itemize}
Review of the [[formal deformation quantization]] of Lie-Poisson structures via transfer of the product on the [[universal enveloping algebra]] of the given Lie algebra is for instance in

\begin{itemize}%
\item Simone Gutt, section 2.2. of \emph{Deformation quantization of Poisson manifolds}, Geometry and Topology Monographs 17 (2011) 171-220 (\href{http://msp.org/gtm/2011/17/gtm-2011-17-003p.pdf}{pdf})

\end{itemize}
and generalization to more general polynomial Poisson algebras is discussed in

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

\end{itemize}
The [[strict deformation quantization]] of Lie-Poisson structures was considered in

\begin{itemize}%
\item [[Marc Rieffel]], \emph{Lie group convolution algebras as deformation quantization of linear Poisson structures}, American Journal of Mathematics Vol. 112, No. 4 (Aug., 1990), pp. 657-685 (\href{http://www.jstor.org/stable/2374874}{jstor})

\end{itemize}
The [[symplectic Lie groupoid]] [[Lie integration|Lie integrating]] Lie-Poisson structures is discussed as example 4.3 in

\begin{itemize}%
\item [[Henrique Bursztyn]], [[Marius Crainic]], \emph{Dirac structures, momentum maps and quasi-Poisson manifolds} (\href{http://www.preprint.impa.br/FullText/Bursztyn__Fri_Dec_23_11_24_19_BRDT_2005.html/alanfestimpa.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item [[Victor Ginzburg]], [[Alan Weinstein]], \emph{Lie-Poisson structure on some Poisson Lie groups}, Journal of the AMS, volume 5, number 2 (1992) (\href{http://www.ams.org/journals/jams/1992-05-02/S0894-0347-1992-1126117-8/S0894-0347-1992-1126117-8.pdf}{pdf})

\end{itemize}
[[!redirects Lie-Poisson structures]]



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