The ascending central series of nilpotent Lie algebras with complex structure

TL;DR

This paper investigates the structure of 8-dimensional nilpotent Lie algebras with complex structures, providing bounds, structural theorems, and explicit descriptions of their ascending central series and complex structure equations.

Contribution

It offers new restrictions on the ascending central series of such Lie algebras and constructs explicit complex structure equations for all cases.

Findings

Bound on the dimension of the center without non-trivial J-invariant ideals

Structural theorem for the ascending central series in 8-dimensional cases

Explicit complex structure equations parametrizing all such pairs

Abstract

We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra $\mathfrak g$ under the presence of a complex structure $J$. In particular, we find a bound for the dimension of the center of $\mathfrak g$ when it does not contain any non-trivial $J$-invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras $\mathfrak g$ admitting this particular type of complex structures $J$. Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs $(\mathfrak g, J)$.