# Classification of three-dimensional complex $\omega$-Lie algebras

### Yin Chen

Northeast Normal University, Changchun, China### Chang Liu

Northeast Normal University, Changchun, China### Runxuan Zhang

Northeast Normal University, Changchun, China

## 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

for all $x,y,z\in L$. The notion of $\omega$-Lie algebras, as a generalization of Lie algebras, was introduced in Nurowski [3]. Fundamental results about finite-dimensional $\omega$-Lie algebras were developed by Zusmanovich [5]. In [3], 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 [3].

## Cite this article

Yin Chen, Chang Liu, Runxuan Zhang, Classification of three-dimensional complex $\omega$-Lie algebras. Port. Math. 71 (2014), no. 2, pp. 97–108

DOI 10.4171/PM/1943