Hypercomplex structures with Hermitian-Norden metrics on four-dimensional Lie algebras

TL;DR

This paper classifies and constructs five types of invariant hypercomplex structures with Hermitian-Norden metrics on four-dimensional Lie groups, analyzing different metric signatures and their geometric properties.

Contribution

It provides explicit constructions of hypercomplex structures with Hermitian-Norden metrics on 4D Lie groups, extending previous classifications to various metric signatures.

Findings

Five types of invariant hypercomplex structures constructed

Different metric signatures analyzed for geometric properties

Extension of previous classifications to Hermitian-Norden metrics

Abstract

Integrable hypercomplex structures with Hermitian and Norden metrics on Lie groups of dimension 4 are considered. The corresponding five types of invariant hypercomplex structures with hyper-Hermitian metric, studied by M.L. Barberis, are constructed here. The different cases regarding the signature of the basic pseudo-Riemannian metric are considered.