Courant algebroid

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In a field of mathematics known as differential geometry, a Courant algebroid is a combination of a Lie algebroid and a quadratic Lie algebra. It was originally introduced in 1997 by Zhang-Ju Liu, Alan Weinstein and Ping Xu to describe the double of a Lie bialgebroid.

1 Definition
2 Properties
3 Examples
4 Skew Symmetric bracket
5 References

[edit] Definition

A Courant algebroid consists of the data a vector bundle E->M with a bracket [.,.]:ΓE x ΓE -> ΓE, a non degenerate inner product $\langle.,.\rangle: E\times E\to \R$ , and a bundle map ρ:E-> TM subject to the following axioms:

[φ,[χ,ψ]] = [[φ,χ],ψ] + [χ,[φ,ψ]]

[φ, f ψ] = ρ(φ) f ψ + f [φ,ψ]

$[\phi,\phi]= \frac12 D\langle \phi,\phi\rangle$

$\rho(\phi)\langle \psi,\psi\rangle= 2\langle [\phi,\psi],\psi\rangle$

where φ,ψ are sections into E and f is a smooth function on the base manifold M. D is the combination $β - 1 ρ T d$ with d the de Rham differential, $ρ T$ the dual map of $ρ$ , and β the map from E to $E *$ induced by the metric.

[edit] Properties

The bracket is not skew-symmetric as one can see from the third axiom. Instead it fulfils a certain Jacobi-identity (first axiom) and a Leibniz rule (second axiom). From these two axioms one can derive that the anchor map ρ is a morphism of brackets:

ρ[φ,ψ] = [ρ(φ),ρ(ψ)].

The fourth rule is an invariance of the inner product under the bracket. Polarization leads to

$\rho(\phi)\langle \chi,\psi\rangle= \langle [\phi,\chi],\psi\rangle +\langle \chi,[\phi,\psi]\rangle .$

[edit] Examples

An example of the Courant algebroid is the Dorfman bracket on the direct sum $TM\oplus T^*M$ , defined as:

$[X+\xi, Y+\eta] = [X,Y]+(\mathcal{L}_X\eta -i(Y) d\xi +i(X)i(Y)H)$

where X,Y are vector fields, ξ,η are 1-forms and H is a closed 3-form twisting the bracket. This bracket is used to describe the integrability of generalized complex structures.

A more general example arises from a Lie algebroid A whose induced differential on $A *$ will be written as d again. Then use the same formula as for the Dorfman bracket with H an A-3-form closed under d.

Another example of a Courant algebroid is a quadratic Lie algebra, i.e. a Lie algebra with an invariant scalar product. Here the base manifold is just a point and thus the anchor map (and D) are trivial.

The example described in the paper by Weinstein et al comes from a Lie bialgebroid, i.e. A a Lie algebroid (with anchor $ρ A$ and bracket $[.,.] A$ ), also its dual $A *$ a Lie algebroid (inducing the differential $d_{A^*}$ on $Λ * A$ ) and $d_{A^*}[X,Y]_A=[d_{A^*}X,Y]_A-[X,d_{A^*}Y]_A$ (where on the RHS you extend the A-bracket to $Λ * A$ using graded Leibniz rule). This notion is symmetric in A and $A *$ (see Roytenberg). Here $E=A\oplus A^*$ with anchor $\rho(X+\alpha)=\rho_A(X)+\rho_{A^*}(\alpha)$ and the bracket is the skew-symmetrization of the above in X and α (equivalently in Y and β):

$[X+\alpha,Y+\beta]= ([X,Y]_A +\mathcal{L}^{A^*}_{\alpha}Y-i_\beta d_{A^*}X) +([\alpha,\beta]_{A^*} +\mathcal{L}^A_X\beta-i_Yd_{A}\alpha)$

[edit] Skew Symmetric bracket

Instead of the definition above one can introduce a skew symmetric bracket as $[[\phi,\psi]]= \frac12\big([\phi,\psi]-[\psi,\phi]\big)$ . This fulfils a homotopic Jacobi-identity.

$[[\phi,[[\psi,\chi]]\,]] +\mathrm{cycl.} = DT(\phi,\psi,\chi)$

where T is $T(\phi,\psi,\chi)=\frac13\langle [\phi,\psi],\chi\rangle +\mathrm{cycl.}$ .

The Leibniz rule and the invariance of the scalar product become modified by the relation $[[\phi,\psi]] = [\phi,\psi] -\frac12 D\langle \phi,\psi\rangle$ and the violation of skew symmetry gets replaced by the axiom