∞-Lie theory (higher geometry)
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.
Let $\pi \in \Gamma(T X) \wedge \Gamma(T X)$ 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 $\mathfrak{P}(X,\pi)$ corresponding to $\pi$ is the cotangent bundle
equipped with the anchor map that sends a differential 1-form $\alpha$ to the vector obtained by contraction with the Poisson bivector $\pi \colon \alpha \mapsto \pi(\alpha,-)$.
The Lie bracket $[-,-] : \Gamma(T^* X) \wedge \Gamma(T^* X) \to \Gamma(T^* X)$ is given by
where $\mathcal{L}$ denotes the Lie derivative and $d_{dR}$ the de Rham differential. This is the unique Lie algebroid bracket on $T^* X \stackrel{\pi}{\to} T X$ which is given on exact differential 1-forms by
for all $f,g \in C^\infty(X)$. On a coordinate patch this reduces to
for $\{x^i\}$ the coordinate functios and $\{\pi^{i j}\}$ the components of the Poisson tensor in these coordinates.
We describe the Chevalley-Eilenberg algebra of the Poisson Lie algebra given by $\pi$, which defines it dually.
Notice that $\pi$ is an element of degree 2 in the exterior algebra $\wedge^\bullet \Gamma(T X)$ of multivector fields on $X$. The Lie bracket on tangent vectors in $\Gamma(T X)$ extends to a bracket $[-,-]_{Sch}$ on multivector field, the Schouten bracket. The defining property of the Poisson structure $\pi$ is that
This makes
into a differential of degree +1 on multivector fields, that squares to 0. We write $CE(\mathfrak{P}(X,\pi))$ for the exterior algebra equipped with this differential.
More explicitly, let $\{x^i\} : X \to \mathbb{R}^{dim X}$ be a coordinate patch. Then the differential of $CE(\mathfrak{P}(X,\pi))$ is given by
We discuss aspects of the ∞-Lie algebroid cohomology of Poisson Lie algebroids $\mathfrak{P}(X,\pi)$. 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 $\{x^i\}$ for $X$.
Then the Chevalley-Eilenberg algebra $CE(\mathfrak{P}(X,\pi))$ is generated from the $x^i$ and the $\partial_i$, and the Weil algebra $W(\mathfrak{P}(X,\pi))$ is generated from $x^i$, $\partial_i$ and their shifted partners, which we shall write $\mathbf{d} x^i$ and $\mathbf{d}\partial_i$. The differential on the Weil algebra we may then write
Notice that $\pi \in CE(\mathfrak{P}(X,\pi))$ is a Lie algebroid cocycle, since
The invariant polynomial in transgression with $\pi$ is
One checks that the following is a Chern-Simons element (see there for more) exhibiting the transgression
in that $d_{W(\mathfrak{P}(X,\pi))} cs_\pi = \omega$, and the restriction of $cs_\pi$ to $CE(\mathfrak{P}(X,\pi))$ is evidently the Poisson tensor $\pi$.
For the record (and for the signs) here is the explicit computation
The invariant polynomial $\omega$ makes $\mathfrak{P}(X,\pi)$ 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 $\Sigma$ with values in a Poisson Lie algebroid on $X$ 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 appendix A of