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

$$\begin{array}{ccccc}{T}^{*}X& & \stackrel{\pi (-)}{\to }& & TX\\ & \searrow & & \swarrow \\ & & X\end{array}$$\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 $\pi :\alpha \mapsto \pi (\alpha ,-)$.

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

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

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

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

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

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

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

This makes

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

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

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

$$d_{𝔓(X,\pi )}:{\partial }_i\mapsto ....$$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 $𝔓(X,\pi )$. 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 $\mathrm{CE}(𝔓(X,\pi ))$ is generated from the $x^i$ and the ${\partial }_i$, and the Weil algebra $W(𝔓(X,\pi ))$ 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

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

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

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

Proposition

The invariant polynomial in transgression with $\pi$ is

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

$${\mathrm{cs}}_{\pi }={\pi }^{ij}{\partial }_i\wedge {\partial }_j+{\partial }_i\wedge d{x}^i\in W(𝔓(X,\pi ))$$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 $d_{W(𝔓(X,\pi ))}{\mathrm{cs}}_{\pi }=\omega$, and the restriction of ${\mathrm{cs}}_{\pi }$ to $\mathrm{CE}(𝔓(X,\pi ))$ is evidently the Poisson tensor $\pi$.

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

$$\begin{array}{rl}{d}_{W(𝔓(X,\pi ))}({\pi }^{ij}{\partial }_i\wedge {\partial }_j+{\partial }_i\wedge d{x}^i)=& d{x}^k({\partial }_k{\pi }^{ij}){\partial }_i\wedge {\partial }_j\\ & +2{\pi }^{ij}(d{\partial }_i)\wedge {\partial }_j\\ & -({\partial }_i{\pi }^{jk}){\partial }_j\wedge {\partial }_k\wedge d{x}^i\\ & +(d{\partial }_i)\wedge (d{x}^i)\\ & +(-)(-)2{\pi }^{ij}{\partial }_i\wedge d{\partial }_j\\ =& (d{\partial }_i)\wedge (d{x}^i)\end{array}.$$\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 $𝔓(X,\pi )$ 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

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

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

$${\mathrm{CS}}_{\omega }(X,\eta )={\eta }_i\wedge d_{\mathrm{dR}}{X}^i+{\pi }^{ij}{\eta }_i\wedge {\eta }_j.$$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)

References

One of the earliest reference seems to be

• Ted Courant, Tangent Lie algebroid (pdf)

A review is for instance in appendix A of

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds (arXiv)