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 sl 2 ( C ) . Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on g .