On rigid 3-dimensional Hom-Lie algebras

In this work we obtain all rigid complex 3-dimensional multiplicative Hom-Lie algebras. This is done by studying all deformations of multiplicative Hom-Lie algebras which are also Lie algebras. As a byproduct, we obtain the well-known classification of 3-dimensional multiplicative (non-Lie) Hom-Lie algebras.