nLab
Poisson Lie algebroid

Contents

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 πΓ(TX)Γ(TX) be a Poisson structure on X, regarded as a bivector.

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

[π,π] Sch=0.[\pi,\pi]_{Sch} = 0 \,.

This makes

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

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

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

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

References

One of the earliest reference seems to be

  • Ted Courant?, Tangent Lie algebroid (pdf)

A review is for instance in appendix A of