LA-Courant algebroids and their applications

TL;DR

This thesis introduces LA-Courant algebroids as infinitesimal counterparts to multiplicative Courant algebroids, exploring their applications in integrating q-Poisson structures and reducing Courant algebroids.

Contribution

It develops the theory of LA-Courant algebroids and introduces pseudo-Dirac structures, expanding the understanding of Courant algebroids and their applications.

Findings

LA-Courant algebroids generalize multiplicative Courant algebroids.

Pseudo-Dirac structures include non-Lagrangian subbundles with Lie algebroid structures.

Applications to q-Poisson structures and Courant algebroid reduction.

Abstract

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.