A monoidal structure on the category of relative Hopf modules

TL;DR

This paper establishes a monoidal structure on the category of relative Hopf modules within a braided monoidal category, contingent on specific algebraic conditions on the involved bialgebra and comodule algebra.

Contribution

It characterizes when the category of relative Hopf modules admits a monoidal structure, linking it to the bialgebra structure of A in Yetter-Drinfeld modules.

Findings

Monoidal structure exists iff A is a bialgebra in Yetter-Drinfeld modules

Defines a right A-action on tensor products of relative Hopf modules

Provides examples illustrating the theoretical framework

Abstract

Let $B$ be a bialgebra, and $A$ a left $B$-comodule algebra in a braided monoidal category $\Cc$, and assume that $A$ is also a coalgebra, with a not-necessarily associative or unital left $B$-action. Then we can define a right $A$-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if $A$ is a bialgebra in the category of left Yetter-Drinfeld modules over $B$. Some examples are given.