Abelian complex structures on solvable Lie algebras

TL;DR

This paper characterizes real solvable Lie algebras with abelian complex structures using affine Lie algebras associated with commutative algebras, revealing that all 4-dimensional cases are central extensions of these affine structures.

Contribution

It introduces a new characterization of solvable Lie algebras with abelian complex structures via affine Lie algebras linked to commutative algebras.

Findings

All 4-dimensional Lie algebras with abelian complex structures are central extensions of affine Lie algebras.

Affine Lie algebras generalize the structure of $rak a rak f rak f (C)$.

Lie groups corresponding to these algebras are complex affine manifolds.

Abstract

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie groups are complex affine manifolds. It turns out that all 4-dimensional Lie algebras carrying abelian complex structures are central extensions of such affine Lie algebras.