Dirac and Generalized Complex Structures on Cartan Geometries

TL;DR

This paper introduces linear Dirac and generalized complex structures on Cartan geometries, establishing criteria for Dirac subalgebras and a bijection with structures on homogeneous spaces.

Contribution

It provides new criteria for Dirac structures on Cartan geometries and links generalized structures to homogeneous spaces.

Findings

Criteria for Dirac subalgebras on Cartan geometries

Bijection between structures on torsion-free Cartan geometries and homogeneous spaces

Framework for analyzing generalized complex structures in geometric contexts

Abstract

We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the equivariant generalized structures on its corresponding homogeneous space.