  In this thesis we develop the notion of LA-Courant algebroids, the
infinitesimal analogue of multiplicative Courant algebroids. Specific
applications include the integration of q- Poisson (d, g)-structures, and the
reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac
structures, (possibly non-Lagrangian) subbundles W \subseteq E of a Courant
algebroid such that the Courant bracket endows W naturally with the structure
of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the
theory of q-Poisson (d, g)-structures.
