Classification of three dimensional complex omega-Lie algebras

TL;DR

This paper classifies all three-dimensional complex omega-Lie algebras, extending previous real classifications and providing a new proof for the complex case, thereby enriching the understanding of these algebraic structures.

Contribution

It offers a comprehensive classification of three-dimensional complex omega-Lie algebras and introduces a novel proof method for this classification.

Findings

Complete classification of three-dimensional complex omega-Lie algebras.

New proof technique for the classification.

Clarification of differences between real and complex cases.

Abstract

A complex $\omega$-Lie algebra is a vector space $L$ over the complex field, equipped with a skew symmetric bracket $[-,-]$ and a bilinear form $\omega$ such that $$[[x,y],z]+[[y,z],x]+ [[z,x],y]=\omega(x,y)z+\omega(y,z)x+\omega(z,x)y$$ for all $x,y,z\in L$. The notion of $\omega$-Lie algebras, as a generalization of Lie algebras, was introduced in Nurowski \cite{Nur2007}. Fundamental results about finite-dimensional $\omega$-Lie algebras were developed by Zusmanovich \cite{Zus2010}. In \cite{Nur2007}, all three dimensional non-Lie real $\omega$-Lie algebras were classified. The purpose of this note is to provide an approach to classify all three dimensional non-Lie complex $\omega$-Lie algebras. Our method also gives a new proof of the classification in Nurowski \cite{Nur2007}.