Versal Deformations of Three Dimensional Lie Algebras as L-infinity Algebras

TL;DR

This paper explores the classification and deformation theory of three-dimensional Lie algebras by treating them as L-infinity algebras, providing a new derivation and understanding of their moduli space.

Contribution

It offers a novel approach to classify three-dimensional Lie algebras via L-infinity algebra deformation theory, including a complete construction of versal deformations.

Findings

Derived classification of 3D Lie algebras using L-infinity framework

Constructed versal deformations explicitly

Characterized the moduli space of Lie algebras

Abstract

We consider versal deformations of 0|3-dimensional L-infinity algebras, which correspond precisely to ordinary (non-graded) three dimensional Lie algebras. The classification of such algebras over C is well known, although we shall give a derivation of this classification using an approach of treating them as L-infinity algebras. Because the symmetric algebra of a three dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.