In a field of mathematics known as differential geometry, a Courant geometry 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, 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 with a bracket, a non degenerate fiber-wise inner product, and a bundle map subject to the following axioms, where φ,ψ,χ are sections of E and f is a smooth function on the base manifoldM. D is the combination with d the de Rham differential, the dual map of, and κ the map from E to induced by the inner product.
Skew-Symmetric Definition
An alternative definition can be given to make the bracket skew-symmetric as This no longer satisfies the Jacobi-identity axiom above. It instead fulfills a homotopic Jacobi-identity. where T is The Leibniz rule and the invariance of the scalar product become modified by the relation and the violation of skew-symmetry gets replaced by the axiom The skew-symmetric bracket together with the derivation D and the Jacobiator T form a strongly homotopic Lie algebra.
Properties
The bracket is not skew-symmetric as one can see from the third axiom. Instead it fulfills a certain Jacobi-identity and a Leibniz rule. From these two axioms one can derive that the anchor map ρ is a morphism of brackets: The fourth rule is an invariance of the inner product under the bracket. Polarization leads to
Examples
An example of the Courant algebroid is the Dorfman bracket on the direct sum with a twist introduced by Ševera, defined as: 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 algebroidA whose induced differential on will be written as d again. Then use the same formula as for the Dorfman bracket with H an A-3-form closed underd. 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 are trivial. The example described in the paper by Weinstein et al. comes from a Lie bialgebroid, i.e. A a Lie algebroid, also its dual a Lie algebroid and . This notion is symmetric in A and . Here with anchor and the bracket is the skew-symmetrization of the above in X and α :
Dirac structures
Given a Courant algebroid with the inner product of split signature, then a Dirac structure is a maximally isotropic integrable vector subbundle L → M, i.e.
Examples
As discovered by Courant and parallel by Dorfman, the graph of a 2-form ω ∈ Ω2 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 whose graph is maximally isotropic and integrable iff = 0, i.e. Π is a Poisson bivector on M.
