Universal Enveloping Algebras and Poincaré-Birkhoff-Witt Theorem for Involutive Hom-Lie Algebras

摘要

Hom-type algebras, in particular Hom-Lie algebras, have attracted quite much attention in recent years. A Hom-Lie algebra is called involutive if its Horn map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Horn-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 generalize the well-known Poincare- Birkhoff-Witt theorem for enveloping algebras of Lie algebras to involutive Hom-Lie algebras.