The moduli space of complex 5-dimensional Lie algebras

TL;DR

This paper explores the structure of the moduli space of complex 5-dimensional Lie algebras, revealing its stratification into orbifolds connected by jump deformations, using versal deformations to understand local neighborhoods.

Contribution

It introduces a stratification of the moduli space into orbifolds via versal deformations, providing a detailed geometric understanding of the space of all such Lie algebras.

Findings

Moduli space stratified into orbifolds by group actions

Identification of jump deformations connecting strata

Construction of versal deformations for local analysis

Abstract

In this paper, we study the moduli space of all complex 5-dimensional Lie algebras, realizing it as a stratification by orbifolds, which are connected by jump deformations. The orbifolds are given by the action of finite groups on very simple complex manifolds. Our method of determining the stratification is based on the construction of versal deformations of the Lie algebras, which allow us to identify natural neighborhoods of the elements in the moduli space.