In differential geometry, a field of mathematics, a Courant algebroid is a vector bundle together with an inner product and a compatible bracket more general than that of a Lie algebroid.
It is named after Theodore Courant, who had implicitly devised in 1990[1] 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.
The general notion of Courant algebroid was introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997.
[2] A Courant algebroid consists of the data a vector bundle
(called anchor) subject to the following axioms: where
is a smooth function on the base manifold
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.
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
form a strongly homotopic Lie algebra.
From these two axioms one can derive that the anchor map
Polarization leads to An example of the Courant algebroid is given by the Dorfman bracket[3] on the direct sum
with a twist introduced by Ševera in 1988,[4] defined as: where
This bracket is used to describe the integrability of generalized complex structures.
A more general example arises from a Lie algebroid
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
The example described in the paper by Weinstein et al. comes from a Lie bialgebroid: if
is a Lie algebroid (inducing the differential
), a Dirac structure is a maximally isotropic integrable vector subbundle
, i.e. As discovered by Courant and parallel by Dorfman, the graph of a 2-form
, i.e. the 2-form is closed under the de Rham differential, i.e. is a presymplectic structure.
A second class of examples arises from bivectors
whose graph is maximally isotropic and integrable if and only if
Given a Courant algebroid with inner product of split signature, a generalized complex structure
is a Dirac structure in the complexified Courant algebroid with the additional property where
means complex conjugation with respect to the standard complex structure on the complexification.
As studied in detail by Gualtieri,[5] the generalized complex structures permit the study of geometry analogous to complex geometry.