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Poisson Lie algebroid

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Symplectic geometry

Contents

Idea

A Poisson Lie algebroid on a manifold XX is a Lie algebroid on XX naturally defined from and defining the structure of a Poisson manifold on XX.

This is the degree-1 example of a tower of related concepts, described at n-symplectic manifold.

Definition

Let πΓ(TX)Γ(TX)\pi \in \Gamma(T X) \wedge \Gamma(T X) 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,π)\mathfrak{P}(X,\pi) corresponding to π\pi is the cotangent bundle

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

equipped with the anchor map that sends a differential 1-form α\alpha to the vector obtained by contraction with the Poisson bivector π:απ(α,)\pi \colon \alpha \mapsto \pi(\alpha,-).

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

[α,β] π(α)β π(β)αd dR(π(α,β)), [\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 dRd_{dR} the de Rham differential. This is the unique Lie algebroid bracket on T *XπTXT^* X \stackrel{\pi}{\to} T X which is given on exact differential 1-forms by

[d dRf,d dRg]=d dR{f,g} [d_{dR} f, d_{dR} g] = d_{dR} \{f,g\}

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

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

for {x i}\{x^i\} the coordinate functios and {π ij}\{\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 π\pi, which defines it dually.

Notice that π\pi is an element of degree 2 in the exterior algebra Γ(TX)\wedge^\bullet \Gamma(T X) of multivector fields on XX. The Lie bracket on tangent vectors in Γ(TX)\Gamma(T X) extends to a bracket [,] Sch[-,-]_{Sch} on multivector field, the Schouten bracket. The defining property of the Poisson structure π\pi is that

[π,π] Sch=0. [\pi,\pi]_{Sch} = 0 \,.

This makes

d CE(𝔓(X,π)):=[π,]:CE(𝔓(X,π))CE(𝔓(X,π))) 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(𝔓(X,π))CE(\mathfrak{P}(X,\pi)) for the exterior algebra equipped with this differential.

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

d 𝔓(X,π):x i2π ij j d_{\mathfrak{P}(X,\pi)} : x^i \mapsto 2 \pi^{i j} \partial_j
d 𝔓(X,π): i.... 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,π)\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}\{x^i\} for XX.

Then the Chevalley-Eilenberg algebra CE(𝔓(X,π))CE(\mathfrak{P}(X,\pi)) is generated from the x ix^i and the i\partial_i, and the Weil algebra W(𝔓(X,π))W(\mathfrak{P}(X,\pi)) is generated from x ix^i, i\partial_i and their shifted partners, which we shall write dx i\mathbf{d} x^i and d i\mathbf{d}\partial_i. The differential on the Weil algebra we may then write

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

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

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

The invariant polynomial in transgression with π\pi is

ω=(d i)(dx i)W(𝔓(X,π)). \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 π=π ij i j+ idx iW(𝔓(X,π)) 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,π))cs π=ωd_{W(\mathfrak{P}(X,\pi))} cs_\pi = \omega, and the restriction of cs πcs_\pi to CE(𝔓(X,π))CE(\mathfrak{P}(X,\pi)) is evidently the Poisson tensor π\pi.

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

d W(𝔓(X,π))(π ij i j+ idx i)= dx k( kπ ij) i j +2π ij(d i) j ( iπ jk) j kdx i +(d i)(dx i) +()()2π ij id j = (d i)(dx i). \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 ω\omega makes 𝔓(X,π)\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

Ω (Σ)W(𝔓(X,π))(X,η) \Omega^\bullet(\Sigma) \leftarrow W(\mathfrak{P}(X,\pi)) (X,\eta)

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

Ω (Σ)W(𝔓(X,π))(ω,cs ω)W(b 2):CS ω(X,η) \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 ω(X,η)=η id dRX i+π ijη iη 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.

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

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)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
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)

Examples

References

One of the earliest reference seems to be

A review is for instance in appendix A of

Revised on November 26, 2014 17:23:43 by Urs Schreiber (89.204.139.117)