On 3-dimensional complex Hom-Lie algebras

Abstract

We study and classify the 3-dimensional Hom-Lie algebras over C. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space g. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps g→g that turn g into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps g→g are not homomorphisms for their products are exhibited. Examples also arise of non-isomorphic families of Hom-Lie algebras which share, however, a fixed Lie-algebra product on g. In particular, this is the case for the complex simple Lie algebra sl2(C). Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on g.