Deformation von Lie-Algebroiden und Dirac-Strukturen

Abstract

In this diploma thesis we discuss the deformation theory of Lie algebroids and Dirac structures. The first chapter gives a short introduction to Dirac structures on manifolds as introduced by Courant in 1990. We also give some physical applications of Dirac structures. In the second chapter we present the deformation theory of Lie algebroids following a recent work of Crainic and Moerdijk. We discuss the subject from three different points of view and show the equivalence of these different interpretations. In the third chapter we give definitions for smooth and formal deformations of Dirac structures on Courant algebroids. To investigate the formal theory, we write the Courant bracket with the use of the Rothstein super-Poisson bracket as a derived bracket. As the main result we show that the obstruction for extending a given formal deformation of a certain order lies in the third Lie algebroid cohomology of the Lie algebroid given by the undeformed Dirac structure.