Bialgebra Coverings and Transfer of Structure

TL;DR

This paper develops a bicategory framework for bialgebras with coverings, enabling transfer of algebraic formulas and generalizing Nichols' results on bialgebra quotients to Hopf algebras.

Contribution

It introduces a new bicategory of bialgebras with coverings and extends Nichols' theorem to broader classes of bialgebra quotients.

Findings

Established a bicategory of bialgebras with coverings

Provided a method to transfer formulas for primitives and antipodes

Generalized Nichols' theorem on bialgebra quotients

Abstract

We introduce the bicategory of bialgebras with coverings (which can be thought of as coalgebra-indexed families of morphisms), and provide a motivating application to the transfer of formulas for primitives and antipode. Additionally, we study properties of this bicategory and various sub-bicategories, and describe some universal constructions. Finally, we generalize Nichols' result on bialgebra quotients of Hopf algebra, which gives conditions on when the resulting bialgebra quotient is a Hopf algebra.