Let's suppose that we have an algebra $\mathcal{A}$ (I don't really care whether it's a unital one or associative). From studying Lie algebras and some of their generalizations, I am used to be able to consider tensor representations. However, their existence is tied to the fact that such algebras naturally have a comultiplication.
Are there any cases where we cannot form tensor products from representations of algebra and what is the obstruction for existence of comultiplication for them?
The only thing that I understand so far is that these algebras probably ought to have only infinite-dimensional representations, since we can easily multiply finite-dimensional matrices.
EDIT:
My original goal is rather humble: I want to understand why the associator (or 6j-symbols) in the universal (quantum) Lie algebras' representations is nontrivial, but pentagonator turns out to be trivial; then I want to think about how this might be generalized to n-associator conditions (in the context of representation theory). One answer (which I read in Deligne-Milne) is that the axioms for tensor categories are such that associator isomorphism exists (might be trivial, might be not) and the only relations are pentagon eqation (i.e. pentagonator is trivial) and hexagon equation. However, I don't understand the reasoning behind these axioms (for example, once we consider 2-groups, I bet these axioms will be wrong if one considers their reresentations, whatever they may be; so it seems that Deligne-Milne have a specific example in their heads, like finite/Lie groups or Lie algebras).
Now, there is another story, this time about quasi-Hopf algebras (which I read in Drinfeld, but didn't understand much as well), where one basically relaxes some conditions for Hopf algebra, for example, $(\mathrm{id} \otimes \Delta) \Delta = (\Delta \otimes \mathrm{id}) \Delta$ now holds up to an isomorphism, and still gets a rich situation to study from representation theory point of view. This way of thinking seems connected to the part of my original goal, concerning the n-associator. However, first, Drinfeld still has only pentagon to satisfy and, second, I still don't understand what forces associator to be nontrivial. In order to understand associator, seemingly simpler thing, first, I tried to think what might force it to become trivial or forbid its existence at all. Hence the question in the title.