Universal enveloping algebras and Poincar\'e-Birkhoff-Witt theorem for involutive Hom-Lie algebras

A Hom-type algebra is called involutive if its Hom map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Hom-associative algebra on a Hom-module. We then apply this construction to obtain the universal enveloping algebra of an involutive Hom-Lie algebra. Finally we obtain a Poincar'e-Birkhoff-Witt theorem for the enveloping associative algebra of an involutive Hom-Lie algebra.