Zakai equation

In a field of mathematics known as differential geometry, a Courant algebroid is a structure which, in a certain sense, blends the concepts of Lie algebroid and of quadratic Lie algebra. This notion, which plays a fundamental role in the study of Hitchin's generalized complex structures, was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990 the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on $TM\oplus T^{\ast }M$, called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids.

Definition

A Courant algebroid consists of the data a vector bundle $E\to M$ with a bracket $[.,.]:\Gamma E\times \Gamma E\to \Gamma E$, a non degenerate fiber-wise inner product $⟨.,.⟩:E\times E\to M\times ℝ$, and a bundle map $\rho :E\to TM$ subject to the following axioms,

$[\varphi ,[\chi ,\psi ]]=[[\varphi ,\chi ],\psi ]+[\chi ,[\varphi ,\psi ]]$

$[\varphi ,f\psi ]=\rho (\varphi )f\psi +f[\varphi ,\psi ]$

$[\varphi ,\varphi ]=\frac{1}{2}D⟨\varphi ,\varphi ⟩$

$\rho (\varphi )⟨\psi ,\psi ⟩=2⟨[\varphi ,\psi ],\psi ⟩$

where φ,ψ,χ are sections of E and f is a smooth function on the base manifold M. D is the combination ${\kappa }^{-1}{\rho }^{T}d$ with d the de Rham differential, ${\rho }^T$ the dual map of $\rho$, and κ the map from E to $E^{\ast }$ induced by the inner product.

Properties

The bracket is not skew-symmetric as one can see from the third axiom. Instead it fulfills 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:

$\rho [\varphi ,\psi ]=[\rho (\varphi ),\rho (\psi )].$

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

$\rho (\varphi )⟨\chi ,\psi ⟩=⟨[\varphi ,\chi ],\psi ⟩+⟨\chi ,[\varphi ,\psi ]⟩.$

Examples

An example of the Courant algebroid is the Dorfman bracket on the direct sum $TM\oplus T^{\ast }M$ with a twist introduced by Ševera, (1998) defined as:

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

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 $A^{\ast }$ 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 ${\rho }_A$ and bracket $[.,.{]}_A$), also its dual $A^{\ast }$ a Lie algebroid (inducing the differential $d_{A^{\ast }}$ on ${\wedge }^{\ast }A$) and $d_{A^{\ast }}[X,Y{]}_A=[d_{A^{\ast }}X,Y{]}_A+[X,d_{A^{\ast }}Y{]}_A$ (where on the RHS you extend the A-bracket to ${\wedge }^{\ast }A$ using graded Leibniz rule). This notion is symmetric in A and $A^{\ast }$ (see Roytenberg). Here $E=A\oplus A^{\ast }$ with anchor $\rho (X+\alpha )={\rho }_A(X)+{\rho }_{A^{\ast }}(\alpha )$ and the bracket is the skew-symmetrization of the above in X and α (equivalently in Y and β):

$[X+\alpha ,Y+\beta ]=([X,Y{]}_A+{ℒ}_{\alpha }^{A^{\ast }}Y-i_{\beta }{d}_{A^{\ast }}X)+([\alpha ,\beta {]}_{A^{\ast }}+{ℒ}_X^A\beta -i_Y{d}_A\alpha )$

Skew-symmetric bracket

Instead of the definition above one can introduce a skew-symmetric bracket as

$[[\varphi ,\psi ]]=\frac{1}{2}([\varphi ,\psi ]-[\psi ,\varphi ])$

This fulfills a homotopic Jacobi-identity.

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

where T is

$T(\varphi ,\psi ,\chi )=\frac{1}{3}⟨[\varphi ,\psi ],\chi ⟩+\text{cycl.}$

The Leibniz rule and the invariance of the scalar product become modified by the relation $[[\varphi ,\psi ]]=[\varphi ,\psi ]-\frac{1}{2}D⟨\varphi ,\psi ⟩$ and the violation of skew-symmetry gets replaced by the axiom

$\rho \circ D=0$

The skew-symmetric bracket together with the derivation D and the Jacobiator T form a strongly homotopic Lie algebra.

Dirac structures

Given a Courant algebroid with the inner product $⟨.,.⟩$ of split signature (e.g. the standard one $TM\oplus T^{\ast }M$), then a Dirac structure is a maximally isotropic integrable vector subbundle L → M, i.e.

$⟨L,L⟩\equiv 0$,

$\mathrm{r}\mathrm{k}L=\frac{1}{2}\mathrm{r}\mathrm{k}E$,

$[\Gamma L,\Gamma L]\subset \Gamma L$.

Examples

As discovered by Courant and parallel by Dorfman, the graph of a 2-form ω ∈ Ω²(M) is maximally isotropic and moreover integrable iff dω = 0, i.e. the 2-form is closed under the de Rham differential, i.e. a presymplectic structure.

A second class of examples arises from bivectors $\Pi \in \Gamma ({\wedge }^{2}TM)$ whose graph is maximally isotropic and integrable iff [Π,Π] = 0, i.e. Π is a Poisson bivector on M.

Generalized complex structures

(see also the main article generalized complex geometry)

Given a Courant algebroid with inner product of split signature. A generalized complex structure L → M is a Dirac structure in the complexified Courant algebroid with the additional property

$L\cap \overline{L}=0$

where $\overline{}$ means complex conjugation with respect to the standard complex structure on the complexification.

As studied in detail by Gualtieri the generalized complex structures permit the study of geometry analogous to complex geometry.

Examples

Examples are beside presymplectic and Poisson structures also the graph of a complex structure J: TM → TM.

References

• Dmitry Roytenberg: Courant algebroids, derived brackets, and even symplectic supermanifolds, PhD thesis Univ. of California Berkeley (1999)