This is the degree-1 example of a tower of related concepts, described at n-symplectic manifold.
The Lie bracket is given by
for all . On a coordinate patch this reduces to
for the coordinate functios and the components of the Poisson tensor in these coordinates.
We describe the Chevalley-Eilenberg algebra of the Poisson Lie algebra given by , which defines it dually.
Notice that is an element of degree 2 in the exterior algebra of multivector fields on . The Lie bracket on tangent vectors in extends to a bracket on multivector field, the Schouten bracket. The defining property of the Poisson structure is that
into a differential of degree +1 on multivector fields, that squares to 0. We write for the exterior algebra equipped with this differential.
More explicitly, let be a coordinate patch. Then the differential of is given by
We shall be lazy (and follow tradition) and write the following formulas in a local coordinate patch for .
Then the Chevalley-Eilenberg algebra is generated from the and the , and the Weil algebra is generated from , and their shifted partners, which we shall write and . The differential on the Weil algebra we may then write
Notice that is a Lie algebroid cocycle, since
The invariant polynomial in transgression with is
One checks that the following is a Chern-Simons element (see there for more) exhibiting the transgression
in that , and the restriction of to is evidently the Poisson tensor .
For the record (and for the signs) here is the explicit computation
The invariant polynomial makes a symplectic ∞-Lie algebroid.
it sends ∞-Lie algebroid valued forms
on a 2-dimensional manifold with values in a Poisson Lie algebroid on to the integral of the Chern-Simons 2-form
which, by the above, is in components
There is a formulation of Legendre transformation in terms of Lie algebroid.
Hopf algebroid (appears as a deformation quantization of a Poisson-Lie algebroid)
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||discussed in:|
|0||symplectic manifold||symplectic manifold||symplectic geometry||Lagrangian submanifold||–||ordinary space of states (in geometric quantization)||geometric quantization|
|1||Poisson Lie algebroid||symplectic groupoid||2-plectic geometry||Poisson sigma-model||coisotropic submanifold (of underlying Poisson manifold)||brane of Poisson sigma-model||2-module = category of modules over strict deformation quantiized algebra of observables||extended geometric quantization of 2d Chern-Simons theory|
|2||Courant Lie 2-algebroid||symplectic 2-groupoid||3-plectic geometry||Courant sigma-model||Dirac structure||D-brane in type II geometry|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
One of the earliest reference seems to be
A review is for instance in appendix A of