Let $A$ be an algebra over a field $k$. Let's say a tensor structure for modules over $A$ is any functorial assignment of an $A$-module structure to $M\otimes_kM'$ for $A$-modules $M, M'$. A good way to construct these is by looking at maps of algebras $A \to A\otimes_k A$. Better yet, we could require tensor structures to be associative, and construct tensor structures via coassociative maps.
If $A = k[G]$ is the algebra of a group $G$, the 'diagonal' (usual) tensor structure for $\operatorname{rep}G$ comes from the Hopf algebra map $\Delta : A \to A\otimes A$ which assumes each element of $G$ is grouplike. Studying $\operatorname{rep}G$ by looking for connections between different tensor structures is interesting in its own right, the case of $G = C_p$, $\operatorname{char}k = p$ has some nice combinatorial results, but I want to move on to schemes.
Let $G$ be a finite group scheme over $k$, and let $k[G]$ its coordinate algebra. It seems, to construct tensor structures for $\operatorname{rep}G$, we should analogously be looking at maps $$A[G(A)] \to A[G(A)]\otimes_A A[G(A)]$$ of $A$-algebras which are naturally coassociative for each $k$-algebra $A$. The usual tensor structure is similarly the one coming from assuming each element of the group $G(A)$ is grouplike. That being said, I'd still prefer to begin by looking at the various coassociative maps $k[G] \to k[G]\otimes k[G]$ that exist on the coordinate algebra, not on the group algebra $k[G(k)]$. Working by example this seems plausible. I have associated some natural maps on $A[G(A)]$ to various maps on $k[G]$, but I'm having trouble formalizing what it is I'm actually doing in a way that generalizes. Informally, it seems that I have things backwards, as $k[G] \to k[G]\times k[G]$ is in the $G\times G \to G$ direction while $A[G(A)]\to A[G(A)]\otimes A[G(A)]$ is in the $G\to G\times G$ direction.
Questions: Is there a method to generating tensor structure via natural maps, beginning with any coassociative $k[G] \to k[G] \otimes_k k[G],$ such that the Hopf algebra map yields the usual tensor structure? Am I missing a better way altogether for constructing tensor-structures on group schemes? Is there are way to rephrase tensor structures for finite groups in terms of comodules that could help me generalize? Also, any reference on these 'tensor structures' generated by coassociative maps is welcome, I'm unsure what else they can be called.
