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

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 \[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].