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].