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, $\mathcal{L}_{ J_{\tiny{\mbox{cn}}} }(\mathbb{R}^{2n}) $, with particular attention to their degenerations and deformations. To this end, we identify certain invariants that remain well-behaved under degenerations while preserving the complex structure. These concepts are then applied to the four-dimensional case. Additionally, we explore applications to the study of left-invariant Hermitian structures on Lie groups, and we discuss some aspects of the deformation theory within $ \mathcal{L}_{ J_{\tiny{\mbox{cn}}} }(\mathbb{R}^{2n}) $.