∞-Lie theory (higher geometry)
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 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 linear Poisson structures.
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
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
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:
This Lie-Poisson structure may be thought of as the unique smooth extension of this bracket on linear functions to all smooth functions on $\mathfrak{g}^\ast$.
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
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
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
(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
The Schouten bracket on multivector fields is given on linear basis elements by
(the canonical commutation relations) and extended as a graded derivation in both arguments.
See at deformation quantization the section Relation to universal enveloping algebras.
The symplectic groupoid integrating the Lie-Poisson structure on $\mathfrak{g}^*$ is the action groupoid $\mathfrak{g}^* //G$ of the coadjoint action. For more see at symplectic groupoid in the section Examples – Of Lie-Poisson stucture.
The symplectic leaves of the Lie-Poisson structure on $\mathfrak{g}^*$ are the coadjoint orbits.
We consider the Poisson Lie algebroid $\mathfrak{P}(\mathfrak{g}^\ast)$ of a Lie-Poisson structure and the Lie algebroid cohomology.
By the discussion at Poisson Lie algebroid, the graded algebra of multivector fields equipped with the differential given by the Schouten bracket with the Poisson bivector
is the Chevalley-Eilenberg algebra of this Lie algebroid:
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
(see also at 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 $(\mathfrak{g},[-,-])$.
But in the simple case of Lie-Poisson structure, this cocycle is in fact exact:
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.
Consider the component-description from above. We show that $x^a \partial_a$ is a coboundary.
First notice that
and
From this we get
Under addition a Lie-Poisson manifold becomes a Poisson Lie group, see there for more.
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
Izu Vaisman, section 3.1 of Lectures on the Geometry of Poisson Manifolds, Birkhäuser 1994
Camille Laurent-Gengoux, Linear Poisson Structures and Lie Algebras, chapter 7 pp 179-203 of Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke (eds.) Poisson Structures, Grundlehren der mathematischen Wissenschaften book series (GL, volume 347) (web)
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
and generalization to more general polynomial Poisson algebras is discussed in
The strict deformation quantization of Lie-Poisson structures was considered in
The symplectic Lie groupoid Lie integrating Lie-Poisson structures is discussed as example 4.3 in
See also