Complex Structures on Stratified Lie Algebras

Abstract

We investigate some properties of complex structures on Lie algebras. In particular, we focus on nilpotent complex structures that are characterised by suitable J-invariant ascending or descending central series, $\mathfrak {d}^{\,j}$ and $\mathfrak {d}_j$ , respectively. We introduce a new descending series $\mathfrak {p}_j$ and use it to prove a new characterisation of nilpotent complex structures. We also examine 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.