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

TL;DR

This work analyzes the landscape of 4-dimensional Lie algebras equipped with left-invariant complex structures, focusing on degenerations and deformations within the family $\mathcal{L}_{ J_{\text{cn}}}(\mathbb{R}^{2n})$. It constructs an invariant toolkit, including GL$(\mathbb{R}^{2n},J_{\text{cn}})$-equivariant maps and Killing-form-type invariants, to detect rigidities and obstructions under degeneration. By systematically cataloguing 38 specific algebras and many parameterized families, the paper maps which complex-structured Lie algebras can degenerate to others and under what conditions, across unimodular and non-unimodular cases, abelian vs non-abelian complex structures. The results illuminate the deformation theory within this geometric-structure space and have implications for the study of left-invariant Hermitian structures on Lie groups, providing explicit degeneration chains and rigidities. Overall, the work advances structural and invariant-based understanding of complex Lie-algebra degenerations, with practical ramifications for complex and Hermitian geometry on Lie groups.

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}) $.