Matched pairs of Courant algebroids

Abstract

Abstract. We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair.

Dirac Actions and Lu’s Lie Algebroid

“…This shows that the element (u * P (µ) m , t * A (µ)| e , 0 m ) is orthogonal to the graph of a P . Since Gr(a P ) ⊥ = Gr(a P ), this proves Equation (21). Taking an inner product of this element with itself, we obtain u * P (µ) m , u * P (µ) m = t * A (µ)| e , t * A |(µ)| e , proving u P (γ P ) = γ g .…”

Holomorphic string algebroids

“…Definition of string algebroids. We start by recalling some basic properties about holomorphic Courant algebroids, following [7,17]. Let X be a complex manifold of dimension n. We denote by O X and C the sheaves of holomorphic functions and C-valued constant functions on X, respectively.…”

“…Here p c 1 (P ) denotes the first Pontryagin class of P with respect to the biinvariant symmetric pairing c on g. Condition (2.11) boils down to the fact that a string algebroid Q has an associated smooth complex Courant algebroid Q⊕T 0,1 X ⊕(T 0,1 X) * [17], combined with the classification results for transitive Courant algebroids in [5,7,32]. A refined version of this obstruction will be considered in Section 3.2.…”

Dirac groupoids and Dirac bialgebroids

“…(2) This construction has some similarities with the one of matched pairs of Courant algebroids in [17]. It would be interesting to understand the relation between the two constructions.…”