Maximal nilpotent complex structures

TL;DR

This paper investigates the properties of nilpotent Lie algebras with nilpotent complex structures, establishing bounds on their invariants and characterizing maximal structures where the complex dimension equals the algebra's step.

Contribution

It proves bounds on the invariant u(J) for nilpotent Lie algebras with complex structures and characterizes maximal nilpotent complex structures where u(J) equals the algebra's dimension.

Findings

u(J) is between 2 and 3 when u( ext{g})=2.

Existence of pairs with u(J) equal to the algebra's dimension for u( ext{g})=3.

A structure theorem for u( ext{g})=3 and MaxN complex structures.

Abstract

Let the pair $(\mathfrak{g},J)$ be a nilpotent Lie algebra $\mathfrak{g}$ (NLA for short) endowed with a nilpotent complex structure $J$. In this paper, motivated by a question in the work of Cordero, Fern\'andez, Gray and Ugarte, we prove that $2\leq \nu(J) \leq 3$ for $(\mathfrak{g},J)$ when $\nu(\mathfrak{g})=2$, where $\nu(\mathfrak{g})$ is the step of $\mathfrak{g}$ and $\nu(J)$ is the unique smallest integer such that $\mathfrak{a}(J)_{\nu(J)}=\mathfrak{g}$ as in Definition 1 and 8 of the paper by Cordero, Fern\'andez, Gray and Ugarte. When $\nu(\mathfrak{g})=3$, for arbitrary $n \geq 3$, there exists a pair $(\mathfrak{g},J)$ such that $\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, for which we call the $J$ in the pair $(\mathfrak{g},J)$, satisfying $\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, a maximal nilpotent (MaxN for short) complex structure. The algebraic dimension of a nilmanifold endowed with a left invariant MaxN complex structure is discussed. Furthermore, a structure theorem is proved for the pair $(\mathfrak{g},J)$, where $\nu(\mathfrak{g})=3$ and $J$ is a MaxN complex structure.