Pseudo-Dirac structures

A Dirac structure is a Lagrangian subbundle of a Courant algebroid, L⊂EL⊂E, which is involutive with respect to the Courant bracket. In particular, LL inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, W⊂EW⊂E, together with a pseudo-connection   on its sections, satisfying a natural integrability condition. As a consequence of the definition, WW will be a Lie algebroid. Allowing non-isotropic subbundles of EE incorporates non-skew tensors and connections into Dirac geometry. Novel examples of pseudo-Dirac structures arise in the context of quasi-Poisson geometry, Lie theory, generalized Kähler geometry, and Dirac Lie groups, among others. Despite their greater generality, we show that pseudo-Dirac structures share many of the key features of Dirac structures. In particular, they behave well under composition with Courant relations.