2019/02/22 by García-Delgado, R., Salgado, G., Sánchez-Valenzuela, O. A.
#17A36 #17B60 #17BXX #FOS: Mathematics #Primary: 17-XX #Rings and Algebras (math.RA) #Secondary: 17A30
paper · doi:10.48550/arxiv.1902.08569
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 \mathfrakg. 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 \mathfrakg → \mathfrakg 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 \mathfrakg → \mathfrakg are not homomorphisms for their products, are exhibited. Examples also arise of non-isomorphic families of HomLie algebras which share, however, a fixed Lie-algebra product on \mathfrakg. In particular, this is the case for the complex simple Lie algebra \mathfraksl2(ℂ). Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on \mathfrakg.