JournalspmVol. 71, No. 2pp. 97–108

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

  • Yin Chen

    Northeast Normal University, Changchun, China
  • Chang Liu

    Northeast Normal University, Changchun, China
  • Runxuan Zhang

    Northeast Normal University, Changchun, China
Classification of three-dimensional complex $\omega$-Lie algebras cover
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Abstract

A complex ω\omega-Lie algebra is a vector space LL over the complex field, equipped with a skew-symmetric bracket [,][-,-] and a bilinear form ω\omega such that

[[x,y],z]+[[y,z],x]+[[z,x],y]=ω(x,y)z+ω(y,z)x+ω(z,x)y[[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,zLx,y,z\in L. The notion of ω\omega-Lie algebras, as a generalization of Lie algebras, was introduced in Nurowski [3]. Fundamental results about finite-dimensional ω\omega-Lie algebras were developed by Zusmanovich [5]. In [3], all three-dimensional non-Lie real ω\omega-Lie algebras were classified. The purpose of this note is to provide an approach to classify all three-dimensional non-Lie complex ω\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 ω\omega-Lie algebras. Port. Math. 71 (2014), no. 2, pp. 97–108

DOI 10.4171/PM/1943