6-dimensional product Lie algebras admitting integrable complex structures

TL;DR

This paper classifies 6-dimensional product Lie algebras that admit integrable complex structures and explores such structures on specific Lie algebras, motivated by geometric structures on manifolds.

Contribution

It provides a classification of 6-dimensional product Lie algebras with integrable complex structures and constructs examples on certain orthogonal Lie algebras.

Findings

Classification of $g imes g$ Lie algebras with integrable complex structures

Construction of complex structures on $o(n) igoplus o(n)$

Relevance to geometric structures on manifolds

Abstract

We classify the 6-dimensional Lie algebras of the form $g\times g$ that admit integrable complex structure. We also endow a Lie algebra of the kind $o(n)\oplus o(n)$ with such a complex structure. The motivation comes from geometric structures \'a la Sasaki on $g$-manifolds.