Cofree Hopf algebras on Hopf bimodule algebras

TL;DR

This paper explores a Hopf algebra structure on cotensor coalgebras related to Hopf bimodule algebras, unifying examples like Clifford algebras and quantum groups, and establishing universal and Rota-Baxter properties.

Contribution

It introduces a new Hopf algebra framework on cotensor coalgebras, demonstrating its universal property and connections to quantum quasi-shuffle and Rota-Baxter algebras.

Findings

Contains universal Clifford algebra and quantum group examples

Shows the algebra is the bosonization of a quantum quasi-shuffle algebra

Establishes a Rota-Baxter algebra structure on the new algebra

Abstract

We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.