Hom-Lie-Hopf algebras

Abstract

We studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general $(\alpha ,\beta )$-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are of $(\alpha ,\mathrm{Id})$-type. We show that the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras form a matched pair of Hom-Hopf algebras. We observe also that, the semi-dualization of a double cross product Hom-Hopf algebra is a bicrossproduct Hom-Hopf algebra. In particular, we apply this result to the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras to obtain Hom-Lie-Hopf algebras.

References in corpus (3)

• Hom-algebras and homology
• Module Hom-algebras
• Lie-Hopf algebras and their Hopf cyclic cohomology