Complex structures on stratified Lie algebras

Abstract

This paper investigates some properties of complex structures on Lie algebras. In particular, we focus on $\text{nilpotent}$ $\text{complex structures}$ that are characterized by a suitable $J$-invariant ascending or descending central series ${\mathfrak{d}}^j$ and ${\mathfrak{d}}_j$ respectively. In this article, we introduce a new descending series ${\mathfrak{p}}_j$ and use it to give proof of a new characterization of nilpotent complex structures. We examine also whether nilpotent complex structures on stratified Lie algebras preserve the strata. We find that there exists a $J$-invariant stratification on a step $2$ nilpotent Lie algebra with a complex structure.

11 pages