# Matrix Lie Groups as 4-Dimensional Hypercomplex Manifolds with Hermitian-Norden metrics

## TL;DR

This paper explores four-dimensional Lie groups as hypercomplex manifolds with Hermitian-Norden metrics, establishing a correspondence between Lie algebra types and matrix representations, and providing explicit examples.

## Contribution

It introduces a classification linking Lie algebra types with matrix representations for hypercomplex structures on 4D Lie groups, including explicit examples.

## Key findings

- Established correspondence between Lie algebra types and matrix representations.
- Constructed explicit examples of hypercomplex structures on known Lie groups.
- Identified integrable hypercomplex structures in four-dimensional Lie groups.

## Abstract

There are studied Lie groups considered as almost hypercomplex Hermitian-Norden manifolds, which are integrable and have the lowest dimension four. It is established a correspondence of the derived Lie algebras of types of invariant hypercomplex structures and the explicit matrix representation of their Lie groups. There are constructed examples of the considered structure of different types on some known Lie groups.