Two examples about zero torsion linear maps on Lie algebras

TL;DR

This paper investigates zero torsion linear maps on non-abelian real Lie algebras, providing examples that show such maps are not always extensions of CR-structures, and describes their equivalence classes.

Contribution

It offers the first explicit examples demonstrating the non-necessity of zero torsion maps being CR-structure extensions and classifies their equivalence classes.

Findings

One example shows zero torsion maps are not CR-structure extensions.

Another example confirms they can be extensions.

Explicit classification of complex structures on g x g.

Abstract

The question of whether or not any zero torsion linear map on a non abelian real Lie algebra g is necessarily an extension of some CR-structure is considered and answered in the negative. Two examples are provided, one in the negative and one in the positive.In both cases, the computation up to equivalence of all zero torsion linear maps on g is used for an explicit description of the equivalence classes of integrable complex structures on the direct product g x g.