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

TL;DR

This paper investigates the space of Lie algebras with invariant complex structures, focusing on their degenerations and deformations, and introduces invariants that remain stable under these processes, with applications to Hermitian structures.

Contribution

It identifies invariants that are stable under degenerations of complex Lie algebras and applies these to classify structures in four dimensions and study their deformations.

Findings

Invariants that remain well-behaved under degenerations.

Classification results for four-dimensional complex Lie algebras.

Insights into deformation theory of complex structures on Lie algebras.

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}) $.