On pseudo-Hermitian quadratic nilpotent Lie algebras

TL;DR

This paper classifies nilpotent Lie algebras with complex and pseudo-Hermitian quadratic structures, providing methods for their construction and a complete classification up to dimension 8.

Contribution

It introduces new construction methods and offers a full classification of nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8.

Findings

Complete classification of nilpotent quadratic Lie algebras with Lorentz-Hermitian metrics

Methods for constructing pseudo-Hermitian quadratic Lie algebras

Inductive description via double extension by planes

Abstract

We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.