Deformations of Courant Algebroids and Dirac Structures via Blended Structures

Abstract

Deformations of a Courant Algebroid E and its Dirac subbundle A have been widely considered under the assumption that the pseudo-Euclidean metric is fixed. In this paper, we attack the same problem in a setting that allows the pseudo-Euclidean metric to deform. Thanks to Roytenberg, a Courant algebroid is equivalent to a symplectic graded Q-manifold of degree 2. From this viewpoint, we extend the notions of graded Q-manifold, DGLA and L_\infty-algebra all to "blended" version so that Poisson manifold, Lie algebroid and Courant algebroid are unified as blended Q-manifolds; and define a submaniold A of "coisotropic type" which naturally generalizes the concepts of coisotropic submanifolds, Lie subalgebroids and Dirac subbundles. It turns out the deformations a blended homological vector field Q is controlled by a blended DGLA, and the deformations of A is controlled by a blended L_\infty-algebra. The results apply to the deformations of a Courant algebroid and its Dirac structures, the deformations of a Poisson manifold and its coisotropic submanifold, the deformations of a Lie algebroid and its Lie subalgebroid.