The aim of this paper is to introduce and study quadratic Hom-Lie algebras,which are Hom-Lie algebras with symmetric invariant nondegenerate bilinearforms. We provide several constructions leading to examples and extend thedouble extension theory to Hom-Lie algebras. We reduce the case where the twistmap is invertible to the study of involutive quadratic Lie algebras. Weestablish a correspondence between the class of involutive quadratic Hom-Liealgebras and quadratic simple Lie algebras with symmetric involution.Centerless involutive quadratic Hom-Lie algebras are characterized. Alsoelements of a representation theory for Hom-Lie algebras, including adjoint andcoadjoint representations are supplied with application to quadratic Hom-Liealgebras.
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