Fix a field $k$, and let $A$ be a (commutative, coassociative, counital) $k$-bialgebra. Write $\otimes = \otimes_k$. The category $\mathrm{Mod}_A$ of $A$-modules admits the structure of a monoidal category, symmetric if $A$ is cocommutative, with underlying tensor product $\otimes$: for $M,N\in\mathrm{Mod}_A$, we have an action on $M\otimes N$ given by
$$ a\cdot (m\otimes n) = \sum_a a_{(1)} m\otimes a_{(2)} n.$$
Here, I use Sweedler's notation $\Delta(a) = \sum_a a_{(1)}\otimes a_{(2)}$ for the coproduct on $A$. From this monoidal structure, we can recover the bialgebra structure on $A$ by
$$\Delta(a) = a\cdot (1\otimes 1),$$
where $1\otimes 1\in A\otimes A$. My question is:
- Suppose given a monoidal structure on $\mathrm{Mod}_A$ with underlying tensor product $\otimes$. When does this arise from the structure of a bialgebra on $A$?
There is also the converse question:
- What coproducts $A\rightarrow A\otimes A$ arise from such monoidal structures on $\mathrm{Mod}_A$?
If there is a good answer to the first question, then I would also be interested in detecting when such a monoidal structure arises from a bialgebra with the property of being a Hopf algebra.
The question boils down to: given a monoidal structure on $\mathrm{Mod}_A$ with underlying monoidal product $\otimes$, what natural conditions ensure that $\Delta(a) = a \cdot (1\otimes 1)$ gives $A$ the structure of a bialgebra? For example, if $A\otimes A$ is an $A-A\otimes A$-bimodule, then $\Delta(ab) = \Delta(a)\Delta(b)$, but it is not clear to me what natural conditions force this to hold and if anything can be said if it does not.