Hom–Lie algebras with symmetric invariant nondegenerate bilinear forms

The aim of this paper is to introduce and study quadratic Hom–Lie algebras, which are Hom–Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom–Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom–Lie algebras. Centerless involutive quadratic Hom–Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom–Lie algebras and quadratic simple Lie algebras with symmetric involution.