Showing changes from revision #22 to #23:
Added | Removed | Changed
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
A Poisson Lie algebroid on a manifold is a Lie algebroid on naturally defined from and defining the structure of a Poisson manifold on .
This is the degree-1 example of a tower of related concepts, described at n-symplectic manifold.
Let be a Poisson manifold structure, incarnated as a Poisson tensor.
In terms of the vector-bundle-with anchor definition of Lie algebroid the Poisson Lie algebroid corresponding to is the cotangent bundle
equipped with the anchor map that sends a differential 1-form to the vector obtained by contraction with the Poisson bivector .
The Lie bracket is given by
where denotes the Lie derivative and the de Rham differential. This is the unique Lie algebroid bracket on which is given on exact differential 1-forms 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
This makes
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 discuss aspects of the ∞-Lie algebroid cohomology of Poisson Lie algebroids . This is equivalently called Poisson cohomology (see there for details).
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.
The infinity-Chern-Simons theory action functional induced from the above Chern-Simons element is that of the Poisson sigma-model:
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
The Lagrangian dg-submanifolds (see there for more) of a Poisson Lie algebroid correspond to the coisotropic submanifolds of the corresponding Poisson manifold.
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.
Poisson Lie algebroid
Hopf algebroid (appears as a deformation quantization of a Poisson-Lie algebroid)
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
One of the earliest reference seems to be
A review is for instance in
The H-cohomology of the graded symplectic form of a Poisson Lie algebroid, regarded a a symplectic Lie n-algebroid, is considered in
Last revised on March 20, 2018 at 22:28:48. See the history of this page for a list of all contributions to it.