Invariant generalized complex structures on Lie groups

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 ( k , ω), where k ⊂𝔤ℂ is an appropriate regular subalgebra of the complex Lie algebra 𝔤ℂ associated to G and ω is a closed 2-form on k , such that Im (ω| k ∩𝔤) is non-degenerate. In the case when G is a semisimple Lie group of inner type (in particular, when G is compact semisimple) 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 is regular, provided that an additional natural condition is satisfied. In the case when G is a semisimple Lie group of outer type, we describe the subalgebras k in terms of appropriate root subsystems of a root system of 𝔤ℂ and we construct a large class of admissible pairs ( k , ω) (hence, regular generalized complex structures on G).