Hom-Lie algebra structures on quadratic Lie algebras and twisted invariant Killing-like forms defined on them

Abstract

Hom-Lie algebras defined on central extensions of a given quadratic Lie algebra that in turn admit an invariant metric, are studied. It is shown how some of these algebras are naturally equipped with other symmetric, bilinear forms that satisfy an invariant condition for their twisted multiplication maps. The twisted invariant bilinear forms so obtained resemble the Cartan-Killing forms defined on ordinary Lie algebras. This fact allows one to reproduce on the Hom-Lie algebras hereby studied, some results that are classically associated to the ordinary Cartan-Killing form.