Invariant generalized complex structures on Lie groups

TL;DR

This paper classifies invariant generalized complex structures on real semisimple Lie groups by reducing the problem to admissible pairs of subalgebras and 2-forms, providing a comprehensive description especially for compact and outer type groups.

Contribution

It introduces the concept of regular invariant generalized complex structures on Lie groups and classifies them for compact and outer type semisimple groups, linking structures to root systems.

Findings

Classification of regular structures on compact semisimple Lie groups

Description of subalgebras via root subsystems for outer type groups

Construction of a large class of admissible pairs (k, \u03c9)

Abstract

We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (\gk, \omega), where \gk is an appropriate regular subalgebra of the complex Lie algebra \gg^{C} associated to G and \omega is a closed 2-form on \gk, such that a non-degeneracy condition holds. In the case when G is a semisimple Lie group of inner type (in particular, when G is compact) a classification of regular generalized complex structures on G is given. We show that any invariant generalized complex structure on a compact semisimple Lie group G is regular, provided that an additional natural condition holds. In the case when G is a semisimple Lie group of outer type, we describe the subalgebras \gk in terms of appropriate root subsystems of a root system of \gg^{C} and we construct a large class of admissible pairs (\gk ,\omega) (hence, regular generalized complex structures on G).