On the variety of Lie algebras endowed with complex structures: Degenerations and deformations

Abstract

We study the space of Lie algebras equipped with left-invariant complex structures, LJcn (R2n),with particular attention to their degenerations and deformations. To this end, we identifycertain invariants that remain well-behaved under degenerations while preserving the complexstructure. These concepts are then applied to the four-dimensional case. Additionally, weexplore applications to the study of left-invariant Hermitian structures on Lie groups, and wediscuss some aspects of the deformation theory within LJcn (R2n).