Complex structures on stratified Lie algebras

TL;DR

This paper explores properties of complex structures on stratified Lie algebras, introducing a new descending series to characterize nilpotent complex structures and examining their preservation of stratification.

Contribution

It introduces a new descending series for nilpotent complex structures and provides a novel characterization, also analyzing their preservation of stratification.

Findings

Existence of a J-invariant stratification on step 2 nilpotent Lie algebras

Introduction of a new descending series for complex structures

Characterization of nilpotent complex structures using the new series

Abstract

This paper investigates some properties of complex structures on Lie algebras. In particular, we focus on $\textit{nilpotent}$ $\textit{complex structures}$ 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.