I really should just start a series called “Yan finally learns simple math” because it takes me ridiculously long to stumble upon ideas that everyone else already knows. Anyway, I have finally convinced myself that I might want to care about Hopf algebras, so maybe this could help someone else.
It is annoying why this realization came so late. I think I’ve asked this question at least five times, each time getting an answer full of words that I didn’t understand. This would be okay, except I think there’s an answer that I think even an undergraduate can appreciate, so I will stick to it now until I become fancy enough to appreciate a “higher” reasoning. Consider some -algebra , and let be an A-module. When we tensor -modules it is obvious that we want to act on them diagonally, so consider . naturally acts on this by the diagonal action .
This is one of those lies that me, being careless, would be willing to buy without thinking much. The source of the lie is that if we’re just multiplying things this is obviously a group homomorphism, but once we allow addition we clearly don’t have enough structure:, which give us terms; we obviously want this to be the same as , and we are stuck with two cross-terms that don’t cancel.
So, we’ve concluded that we can’t do this in general. What happens is that is not naturally an -module; what it is is an -module, in which case we can easily check the obvious map works perfectly.
Well, that would be end of the story, except that there are some algebraic structures that come up often enough where we seem to have this extra power. To be precise, let’s look at the group algebra of some group . Here, for every group element , I just let be the diagonal action, which we can then extend linearly into a -action. Now, this is completely well-defined and quite useful. At first I stared at this for a while wondering why it works, because it seems to be the exact thing we wanted to do earlier that didn’t work. Of course, a little bit of staring gives us the answer – here we had more structure underneath. In particular, we had a group structure that we used and then linearized afterwards, which we did not have the luxury of doing before. Of course, when we study group rings we’re really studying the representations of these groups, so Hopf algebras naturally come up in representation theory.
So, thinking a bit more like a modern mathematician – what we really have here is a nontrivial map , meaning that our natural -action really came from such a map. This is a weird thing that is the opposite of the normal product, which is a map . The structure of the previous sentence makes it retroactively intuitive that this operation, the co-product, is the extra structure we have in these situations. The Hopf algebra is just the formalism to capture these situations.
A similar situation appears in algebraic topology. Why is cohomology nicer than homology? Well, for starters we have a nice product, the cup product, which makes the cohomology ring a… ring. When our space is a Lie group, we have a group structure, and thus a group product. This group product induces a coproduct on the cohomology ring. making it a Hopf algebra as well.
This explains why a Hopf algebra may be a nice definition to have for labeling things. Now, I’m still not sure if they’re immediately useful for elucidating concepts or for doing specific things in combinatorics, but I’ll keep my mind open and see. Right now I’m just confused on why I’ve never managed to understand this before, especially given that the Wikipedia article seems to cover most of what I said; I’ve concluded it may be just because I was too mathematically immature to follow the explainer (I’m including Wikipedia as a possible explainer)’s reasoning at the time, in which case I apologize retroactively to the explainer.
Much thanks to Henry Cohn’s “Quantum Groups” article for making this click in my head, and thanks to Tiankai Liu for figuring this one out with me.