Hom-Lie-Hopf algebras

TL;DR

This paper explores the construction and properties of Hom-Hopf algebras derived from Hom-Lie algebras, focusing on double cross products, bicrossproducts, and their universal enveloping algebras, revealing new structural relationships.

Contribution

It introduces the double cross product and bicrossproduct constructions for Hom-Hopf algebras of general type and studies their applications to universal enveloping Hom-Hopf algebras of Hom-Lie algebras.

Findings

Universal enveloping Hom-Hopf algebras of a matched pair form a matched pair of Hom-Hopf algebras.

Semi-dualization of a double cross product yields a bicrossproduct Hom-Hopf algebra.

Application to Hom-Lie algebras produces new Hom-Lie-Hopf algebra structures.

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 $(\alpha,{\rm Id})$-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.