A monoidal structure on the category of relative Hopf modules

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.

💡 Research Summary

The paper investigates a monoidal structure on the category of relative Hopf modules in a very general setting. Let 𝒞 be a braided monoidal category, B a bialgebra in 𝒞, and A a left B‑comodule algebra which is also a coalgebra. The authors do not require the left B‑action on A to be associative or unital; only a morphism ⊲:B⊗A→A is assumed.

A relative Hopf module (M,ρ_M,·) is defined as a left B‑comodule M equipped with a right A‑action such that the usual Hopf compatibility condition holds: \