Courant algebroids, derived brackets and even symplectic supermanifolds

Abstract

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complicated properties of Courant algebroids, we interpret them by associating to each Courant algebroid a strongly homotopy Lie algebra in a natural way. Next, we propose an alternative construction of the double of a Lie bialgebroid as a homological hamiltonian vector field on an even symplectic supermanifold. The classical BRST complex and the Weil algebra arise as special cases. We recover the Courant algebroid via the derived bracket construction and give a simple proof of the doubling theorem of Liu, Weinstein and Xu. We also introduce a generalization, quasi-Lie bialgebroids, analogous to Drinfeld's quasi-Lie bialgebras; we show that the derived bracket construction in this case also yields a Courant algebroid. Finally, we compute the Poisson cohomology of a one-parameter family of SU(2)- covariant Poisson structures on S^2. As an application, we show that these structures are non-trivial deformations of each other, and that they do not admit rescaling.