nLab Lie-Poisson structure

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 formal 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 $\mathfrak{g}^* \sslash G$ 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

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

(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:

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.

Related concepts

• 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:

• François Bayen, Moshé Flato, Christian Fronsdal, André Lichnerowicz, Daniel Sternheimer, §2 in: Deformation theory and quantization. I. Deformations of symplectic structures., Annals of Physics 111 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

• Alan Weinstein, §3 in: The local structure of Poisson manifolds, J. Differential Geom. 18 3 (1983) 523-557 [doi:10.4310/jdg/1214437787]

• 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 of: Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke (eds.), Poisson Structures, Grundlehren der mathematischen Wissenschaften 347 179-203 [doi:10.1007/978-3-642-31090-4_7]

Monograph:

• Stefan Waldmann, pp. 220 in: Poisson-Geometrie und Deformationsquantisierung, Springer (2007) [doi:10.1007/978-3-540-72518-3]

On the formal deformation quantization of Lie-Poisson structures via transfer of the product on the universal enveloping algebra of the given Lie algebra:

• Simone Gutt, An explicit $\ast$-product on the cotangent bundle of a Lie group, Letters in Mathematical Physics 7 (1983) 249–258 [doi:10.1007/BF00400441]

reviewed in:

• Simone Gutt, §2.2 of: Deformation quantization of Poisson manifolds, Geometry and Topology Monographs 17 (2011) 171-220 [gtm:17, pdf, arXiv:math/0003107]

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)

On strict deformation quantization of Lie-Poisson structures via Lie-group algebras:

• Marc A. Rieffel, Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures, American Journal of Mathematics 112 4 (1990) 657-685 [doi:10.2307/2374874, jstor:2374874]

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

• Henrique Bursztyn, Marius Crainic, Dirac structures, momentum maps and quasi-Poisson manifolds (pdf)

See also

• Victor Ginzburg, Alan Weinstein, Lie-Poisson structure on some Poisson Lie groups, Journal of the AMS, volume 5, number 2 (1992) (pdf)