Extending invariant complex structures

TL;DR

This paper investigates how to extend complex structures from an ideal within a Lie algebra to the entire algebra, considering additional structures like metrics or symplectic forms, and explores the algebraic constraints involved.

Contribution

It provides a systematic analysis of extending complex structures under various conditions and offers explicit examples, especially in six dimensions.

Findings

Constraints on algebraic structure of g when extending complex structures

Explicit examples and computations in six-dimensional Lie algebras

Conditions for compatibility with metrics or symplectic structures

Abstract

We study the problem of extending a complex structure to a given Lie algebra g, which is firstly defined on an ideal h of g. We consider the next situations: h is either complex or it is totally real. The next question is to equip g with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either h is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of g. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.