nLab
Poisson Lie algebroid

Idea

A Poisson Lie algebroid on a manifold $X$ is a Lie algebroid on $X$ naturally defined from and defining the structure of a Poisson manifold on $X$.

This is the degree-1 example of a tower of related concepts, described at n-symplectic manifold.

Definition

Let $\pi \in \Gamma (TX)\wedge \Gamma (TX)$ be a Poisson structure on $X$, regarded as a bivector.

This is an element of degree 2 in the exterior algebra ${\wedge }^{•}\Gamma (TX)$ of multivector fields on $X$. The Lie bracket on tangent vectors in $\Gamma (TX)$ extends to a bracket [-,-]_{Sch on multivector field, the Schouten bracket. The defining property of the Poisson structure $\pi$ is that

$$[\pi ,\pi {]}_{\mathrm{Sch}}=0.$$[\pi,\pi]_{Sch} = 0 \,.

This makes

$$d_{\mathrm{CE}(X,\pi )}:=[\pi ,-]:\mathrm{CE}(X,\pi )\to \mathrm{CE}(X,\pi )$$d_{CE(X,\pi)} := [\pi, -] : CE(X,\pi) \to CE(X,\pi)

into a differential of degree +1 that squares to 0. We write $\mathrm{CE}(X,\pi )$ for the exterior algebra equipped with this differential. Then $\mathrm{CE}(X,\pi )$ is the Chevalley-Eilenberg algebra of some Lie algebroid. This is the Poisson Lie algebroid of $(X,\pi )$.

In terms of the vector-bundle-with anchor definition of Lie algebroid this is

$$\begin{array}{ccccc}{T}^{*}X& & \stackrel{\pi (-)}{\to }& & TX\\ & \searrow & & \swarrow \\ & & X\end{array}.$$\array{ T^* X &&\stackrel{\pi(-)}{\to}&& T X \\ & \searrow && \swarrow \\ && X } \,.

Related concepts

• Under Lie integration a Poisson Lie algebroid is supposed to yield a symplectic groupoid.

• There is a formulation of Legendre transformation in terms of Lie algebroid.

References

One of the earliest reference seems to be

• Ted Courant?, Tangent Lie algebroid (pdf)

A review is for instance in appendix A of

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds (arXiv)