# nLab Lie-Poisson structure

Contents

## Examples

### $\infty$-Lie algebras

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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 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.

## Definition

### 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

$\mathbf{d} f|_{\alpha} \colon T_\alpha \mathfrak{g}^\ast = \mathfrak{g}^\ast \longrightarrow \mathbb{R}$

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

$\{f,g\} \;\colon\; \theta \mapsto -\theta ([\mathbf{d} f, \mathbf{d} g]) \,.$

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:

$\left\{ \langle -, v_1\rangle, \langle -, v_2\rangle, \right\} = \langle - ,[v_1,v_2]\rangle \,.$

This Lie-Poisson structure may be thought of as the unique smooth extension of this bracket on linear functions to all smooth functions on $\mathfrak{g}^\ast$.

### In components

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

$[x^a,x^b] = \underset{c}{\sum} C^{a b}{}_c x^c \,.$

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

$\partial_a x^b = \delta_a^b = \left\{ \array{ 1 & if\; a=b \\ 0 & otherwise } \right.$

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

$v^{a_1 \cdots a_q} \partial_{a_1}\wedge \cdots \wedge \partial_{a_q}$

(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

$\pi = \tfrac{1}{2}\underset{a,b,c}{\sum} C^{a b}{}_c x^c \partial_a \wedge \partial_b \,.$

The Schouten bracket on multivector fields is given on linear basis elements by

$\{x^a, x^b\}_{Sch} = 0$
$\{\partial_a, x^b\}_{Sch} = \delta_a^b$
$\{\partial_a, \partial_b\}_{Sch} = 0$

(the canonical commutation relations) and extended as a graded derivation in both arguments.

## Properties

### Deformation quantization by universal enveloping algebra

See at deformation quantization the section Relation to universal enveloping algebras.

### Symplectic groupoid

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.

### Symplectic leaves

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

### Poisson-Lie algebroid cohomology

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

###### Remark

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

$d_{CE} = \{\pi, -\}_{Sch}$

is the Chevalley-Eilenberg algebra of this Lie algebroid:

$CE(\mathfrak{P}(\mathfrak{g}^\ast)) = \left( \wedge^\bullet \Gamma(T\mathfrak{g}^\ast), d_{CE} = \{\pi, -\}_{Sch} \right) \,.$
###### Remark

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

$d_{CE}\pi = \{\pi,\pi\}_{Sch} = 0$

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:

###### Proposition

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.

###### Proof

Consider the component-description from above. We show that $x^a \partial_a$ is a coboundary.

First notice that

$\{x^b, x^a \partial_a \}_{Sch} = -x^b$

and

$\{\partial_b, x^a \partial_a \}_{Sch} = \partial_b \,.$

From this we get

\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}

### Poisson Lie group structure

Under addition a Lie-Poisson manifold becomes a Poisson Lie group, see there for more.

• a moment map is often expresses as a Poisson homomorphism into a Lie-Poisson structure.

## 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

• 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

• Simone Gutt, section 2.2. of Deformation quantization of Poisson manifolds, Geometry and Topology Monographs 17 (2011) 171-220 (pdf)

and generalization to more general polynomial Poisson algebras is discussed in

• Michael Penkava, Pol Vanhaecke, Deformation Quantization of Polynomial Poisson Algebras, Journal of Algebra 227, 365ñ393 (2000) (arXiv:math/9804022)

The strict deformation quantization of Lie-Poisson structures was considered in

• Marc Rieffel, Lie group convolution algebras as deformation quantization of linear Poisson structures, American Journal of Mathematics

Vol. 112, No. 4 (Aug., 1990), pp. 657-685 (jstor)

The symplectic Lie groupoid Lie integrating Lie-Poisson structures is discussed as example 4.3 in