Classification of four-dimensional Lie algebras admitting a para-hypercomplex structure

TL;DR

This paper classifies 4-dimensional real Lie algebras that admit a para-hypercomplex structure, advancing the understanding of Lie groups with specific invariant geometric structures and metrics.

Contribution

It provides a classification of 4-dimensional Lie algebras with para-hypercomplex structures, extending previous work on hypercomplex structures to the para-hypercomplex case.

Findings

Identifies all 4D Lie algebras with para-hypercomplex structures.

Establishes a connection to Lie groups with anti-self-dual metrics.

Complements existing classifications of hypercomplex structures.

Abstract

The main goal is to classify 4-dimensional real Lie algebras $\g$ which admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-self-dual metric. Our study is related to the work of Barberis who classified real, 4-dimensional simply-connected Lie groups which admit an invariant hypercomplex structure.