# nLab Poisson Lie algebroid

∞-Lie theory

## Examples

### $\infty$-Lie algebras

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# 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 $\pi \in \Gamma \left(TX\right)\wedge \Gamma \left(TX\right)$ 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 $𝔓\left(X,\pi \right)$ corresponding to $\pi$ is the cotangent bundle

$\begin{array}{ccccc}{T}^{*}X& & \stackrel{\pi \left(-\right)}{\to }& & TX\\ & ↘& & ↙\\ & & X\end{array}$\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 $\pi :\alpha ↦\pi \left(\alpha ,-\right)$.

The Lie bracket $\left[-,-\right]:\Gamma \left({T}^{*}X\right)\wedge \Gamma \left({T}^{*}X\right)\to \Gamma \left({T}^{*}X\right)$ is given by

$\left[\alpha ,\beta \right]≔{ℒ}_{\pi \left(\alpha \right)}\beta -{ℒ}_{\pi \left(\beta \right)}\alpha -{d}_{\mathrm{dR}}\left(\pi \left(\alpha ,\beta \right)\right)\phantom{\rule{thinmathspace}{0ex}},$[\alpha,\beta] \coloneqq \mathcal{L}_{\pi(\alpha)} \beta - \mathcal{L}_{\pi(\beta)} \alpha - d_{dR}(\pi(\alpha,\beta))\,,

where $ℒ$ denotes the Lie derivative and ${d}_{\mathrm{dR}}$ the de Rham differential. This is the unique Lie algebroid bracket on ${T}^{*}X\stackrel{\pi }{\to }TX$ which is given on exact differential 1-forms by

$\left[{d}_{\mathrm{dR}}f,{d}_{\mathrm{dR}}g\right]={d}_{\mathrm{dR}}\left\{f,g\right\}$[d_{dR} f, d_{dR} g] = d_{dR} \{f,g\}

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

$\left[d{x}^{i},d{x}^{j}\right]={d}_{\mathrm{dR}}{\pi }^{ij}$[d x^i , d x^j] = d_{dR} \pi^{i j}

for $\left\{{x}^{i}\right\}$ the coordinate functios and $\left\{{\pi }^{ij}\right\}$ 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 ${\wedge }^{•}\Gamma \left(TX\right)$ of multivector fields on $X$. The Lie bracket on tangent vectors in $\Gamma \left(TX\right)$ extends to a bracket $\left[-,-{\right]}_{\mathrm{Sch}}$ on multivector field, the Schouten bracket. The defining property of the Poisson structure $\pi$ is that

$\left[\pi ,\pi {\right]}_{\mathrm{Sch}}=0\phantom{\rule{thinmathspace}{0ex}}.$[\pi,\pi]_{Sch} = 0 \,.

This makes

${d}_{\mathrm{CE}\left(𝔓\left(X,\pi \right)\right)}:=\left[\pi ,-\right]:\mathrm{CE}\left(𝔓\left(X,\pi \right)\right)\to \mathrm{CE}\left(𝔓\left(X,\pi \right)\right)\right)$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 $\mathrm{CE}\left(𝔓\left(X,\pi \right)\right)$ for the exterior algebra equipped with this differential.

More explicitly, let $\left\{{x}^{i}\right\}:X\to {ℝ}^{\mathrm{dim}X}$ be a coordinate patch. Then the differential of $\mathrm{CE}\left(𝔓\left(X,\pi \right)\right)$ is given by

${d}_{𝔓\left(X,\pi \right)}:{x}^{i}↦2{\pi }^{ij}{\partial }_{j}$d_{\mathfrak{P}(X,\pi)} : x^i \mapsto 2 \pi^{i j} \partial_j
${d}_{𝔓\left(X,\pi \right)}:{\partial }_{i}↦...\phantom{\rule{thinmathspace}{0ex}}.$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 $𝔓\left(X,\pi \right)$. 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 $\left\{{x}^{i}\right\}$ for $X$.

Then the Chevalley-Eilenberg algebra $\mathrm{CE}\left(𝔓\left(X,\pi \right)\right)$ is generated from the ${x}^{i}$ and the ${\partial }_{i}$, and the Weil algebra $W\left(𝔓\left(X,\pi \right)\right)$ is generated from ${x}^{i}$, ${\partial }_{i}$ and their shifted partners, which we shall write $d{x}^{i}$ and $d{\partial }_{i}$. The differential on the Weil algebra we may then write

${d}_{W\left(𝔓\left(X,\pi \right)\right)}=\left[\pi ,-{\right]}_{\mathrm{Sch}}+d\phantom{\rule{thinmathspace}{0ex}}.$d_{W(\mathfrak{P}(X,\pi))} = [\pi,-]_{Sch} + \mathbf{d} \,.

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

${d}_{\mathrm{CE}\left(𝔓\left(X,\pi \right)\right)}\pi =\left[\pi ,\pi {\right]}_{\mathrm{Sch}}=0\phantom{\rule{thinmathspace}{0ex}}.$d_{CE(\mathfrak{P}(X,\pi))} \pi = [\pi,\pi]_{Sch} = 0 \,.
###### Proposition

The invariant polynomial in transgression with $\pi$ is

$\omega =\left(d{\partial }_{i}\right)\wedge \left(d{x}^{i}\right)\in W\left(𝔓\left(X,\pi \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

${\mathrm{cs}}_{\pi }={\pi }^{ij}{\partial }_{i}\wedge {\partial }_{j}+{\partial }_{i}\wedge d{x}^{i}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in W\left(𝔓\left(X,\pi \right)\right)$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 ${d}_{W\left(𝔓\left(X,\pi \right)\right)}{\mathrm{cs}}_{\pi }=\omega$, and the restriction of ${\mathrm{cs}}_{\pi }$ to $\mathrm{CE}\left(𝔓\left(X,\pi \right)\right)$ is evidently the Poisson tensor $\pi$.

For the record (and for the signs) here is the explicit computation

$\begin{array}{rl}{d}_{W\left(𝔓\left(X,\pi \right)\right)}\left({\pi }^{ij}{\partial }_{i}\wedge {\partial }_{j}+{\partial }_{i}\wedge d{x}^{i}\right)=& d{x}^{k}\left({\partial }_{k}{\pi }^{ij}\right){\partial }_{i}\wedge {\partial }_{j}\\ & +2{\pi }^{ij}\left(d{\partial }_{i}\right)\wedge {\partial }_{j}\\ & -\left({\partial }_{i}{\pi }^{jk}\right){\partial }_{j}\wedge {\partial }_{k}\wedge d{x}^{i}\\ & +\left(d{\partial }_{i}\right)\wedge \left(d{x}^{i}\right)\\ & +\left(-\right)\left(-\right)2{\pi }^{ij}{\partial }_{i}\wedge d{\partial }_{j}\\ =& \left(d{\partial }_{i}\right)\wedge \left(d{x}^{i}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $𝔓\left(X,\pi \right)$ 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

${\Omega }^{•}\left(\Sigma \right)←W\left(𝔓\left(X,\pi \right)\right)\left(X,\eta \right)$\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

${\Omega }^{•}\left(\Sigma \right)←W\left(𝔓\left(X,\pi \right)\right)\stackrel{\left(\omega ,{\mathrm{cs}}_{\omega }\right)}{←}W\left({b}^{2}ℝ\right):{\mathrm{CS}}_{\omega }\left(X,\eta \right)$\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

${\mathrm{CS}}_{\omega }\left(X,\eta \right)={\eta }_{i}\wedge {d}_{\mathrm{dR}}{X}^{i}+{\pi }^{ij}{\eta }_{i}\wedge {\eta }_{j}\phantom{\rule{thinmathspace}{0ex}}.$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.

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

$n\in ℕ$symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $\left(n+1\right)$-d sigma-modelhigher symplectic geometry$\left(n+1\right)$d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension $\left(n+1\right)$discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
$n$symplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometry$d=n+1$ AKSZ sigma-model