Complex and biHermitian structures on four dimensional real Lie algebras

TL;DR

This paper introduces a new method for classifying complex and biHermitian structures on four-dimensional real Lie algebras by transforming geometric conditions into matrix relations and solving them systematically.

Contribution

It presents a novel approach using non-coordinate basis and automorphism groups to classify structures on low-dimensional Lie algebras.

Findings

Complete classification of complex structures on four-dimensional real Lie algebras

Systematic method applicable to low-dimensional Lie algebras

Matrix relations facilitate the classification process

Abstract

We give a new method for calculation of complex and biHermitian structures on low dimensional real Lie algebras. In this method, using non-coordinate basis, we first transform the Nijenhuis tensor field and biHermitian structure relations on Lie groups to the tensor relations on their Lie algebras. Then we use adjoint representation for writing these relations in the matrix form; in this manner by solving these matrix relations and using automorphism groups of four dimensional real Lie algebras we obtain and classify all complex and biHermitian structures on four dimensional real Lie algebras.