Classification of three dimensional complex omega-Lie algebras

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

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