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 $\pi \in \Gamma(\Wedge^2 TX)$ be a Poisson manifold structure, incarnated as a Poisson tensor.

As vector-bundle with anchor

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

$$\array{ T^* X &&\stackrel{\pi(-)}{\to}&& T X \\ & \searrow && \swarrow \\ && X }$$

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

$$[\alpha,\beta] \coloneqq \mathcal{L}_{\pi(\alpha)} \beta - \mathcal{L}_{\pi(\beta)} \alpha - d_{dR}(\pi(\alpha,\beta))\,,$$

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

$$[d_{dR} f, d_{dR} g] = d_{dR} \{f,g\}$$

for all $f,g \in C^\infty(X)$. On a coordinate patch this reduces to

$$[d x^i , d x^j] = d_{dR} \pi^{i j}$$

for $\{x^i\}$ the coordinate functios and $\{\pi^{i j}\}$ the components of the Poisson tensor in these coordinates.

Chevalley-Eilenberg algebra

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

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

This makes

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

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

$$d_{\mathfrak{P}(X,\pi)} : x^i \mapsto 2 \pi^{i j} \partial_j$$

$$d_{\mathfrak{P}(X,\pi)} : \partial_i \mapsto ... \,.$$

Properties

Cohomology and Chern-Simons elements

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

$$d_{W(\mathfrak{P}(X,\pi))} = [\pi,-]_{Sch} + \mathbf{d} \,.$$

Notice that $\pi \in CE(\mathfrak{P}(X,\pi))$ is a Lie algebroid cocycle, since

$$d_{CE(\mathfrak{P}(X,\pi))} \pi = [\pi,\pi]_{Sch} = 0 \,.$$

Proposition

The invariant polynomial in transgression with $\pi$ is

$$\omega = (\mathbf{d}\partial_i) \wedge (\mathbf{d}x^i) \in W(\mathfrak{P}(X,\pi)) \,.$$

Proof

One checks that the following is a Chern-Simons element (see there for more) exhibiting the transgression

$$cs_\pi = \pi^{i j} \partial_i \wedge \partial_j + \partial_i \wedge \mathbf{d}x^i \;\;\; \in W(\mathfrak{P}(X,\pi))$$

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

$$\begin{aligned} d_{W(\mathfrak{P}(X,\pi))} (\pi^{i j} \partial_i \wedge \partial_j + \partial_i \wedge \mathbf{d} x^i) = & \mathbf{d}x^k (\partial_k \pi^{i j}) \partial_i \wedge \partial_j \\ & + 2 \pi^{i j} (\mathbf{d}\partial_i) \wedge \partial_j \\ & - (\partial_i \pi^{j k}) \partial_j \wedge \partial_k \wedge \mathbf{d}x^i \\ & + (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i) \\ & + (-)(-) 2\pi^{i j} \partial_i \wedge \mathbf{d}\partial_j \\ = & (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i) \end{aligned} \,.$$

Remark

The invariant polynomial $\omega$ makes $\mathfrak{P}(X,\pi)$ a symplectic ∞-Lie algebroid.

Remark

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

$$\Omega^\bullet(\Sigma) \leftarrow W(\mathfrak{P}(X,\pi)) (X,\eta)$$

on a 2-dimensional manifold $\Sigma$ with values in a Poisson Lie algebroid on $X$ to the integral of the Chern-Simons 2-form

$$\Omega^\bullet(\Sigma) \leftarrow W(\mathfrak{P}(X,\pi)) \stackrel{(\omega, cs_\omega)}{\leftarrow} W(b^2 \mathbb{R}) : CS_\omega(X,\eta)$$

which, by the above, is in components

$$CS_\omega(X,\eta) = \eta_i \wedge d_{dR} X^i + \pi^{i j} \eta_i \wedge \eta_j \,.$$

Lagrangian submanifolds and coisotropic submanifolds

The Lagrangian dg-submanifolds (see there for more) of a Poisson Lie algebroid correspond to the coisotropic submanifolds of the corresponding Poisson manifold.

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.

• symplectic Lie n-algebroid

  • symplectic manifold

  • Poisson Lie algebroid

  • Courant algebroid

• Hopf algebroid (appears as a deformation quantization of a Poisson-Lie algebroid)

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	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
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

Examples

• For the simple but important special case of Lie-Poisson structure see there at Poisson-Lie algebroid cohomology.

References

One of the earliest reference seems to be

• Ted Courant, Tangent Lie algebroid (pdf)

A review is for instance in

• Dmitry Roytenberg, appendix A Courant algebroids, derived brackets and even symplectic supermanifolds (arXiv:math/9910078)

The H-cohomology of the graded symplectic form of a Poisson Lie algebroid, regarded a a symplectic Lie n-algebroid, is considered in

• Pavol ?evera?, p. 1 of On the origin of the BV operator on odd symplectic supermanifolds, Lett Math Phys (2006) 78: 55. (arXiv:0506331)