What is...a Hopf algebra?

If I am not mistaken, in the tensor category theory, the notion of Frobenius algebra object generalizes the notion of Hopf algebra object. In the unitary framework, a Frobenius algebra object corresponds to a subfactor, and a Hopf algebra object corresponds to a depth 2 irreducible subfactor.

[@VisualMath]

I thought they were actually quite different things. Hmm, Vect is my favorite tensor category and Frobenius algebra object = Frobenius algebra while Hopf algebra object = Hopf algebra, which are quite different imho. Maybe I am missing something? Very likely! Would be great if you could elaborate.

@@VisualMath I had that in mind but I cannot find the reference (yet). I can still say the following: according to the Wikipedia page on Frobenius algebra (Section Examples, 6.), a finite dimensional Hopf algebra is a Frobenius algebra. Next, by Definition 7.20.3 in EGNO book, to prove that a bialgebra object A is a Frobenius algebra object, we need to show that the comultiplication morphism is a homomorphism of A-bimodule.

[@VisualMath]

@ Very interesting and exciting. For me the "fd Hopf -> Frobenius" was always a weird fluke. In particular, the structure maps change under this identification. Something like "The Frobenius algebra comultiplication on A is that of a Hopf algebra if and only if A=field." is true or something equally strong. In other words, even for K[G] the Frobenius structure is "complicated" compared to the Hopf structure. If you have a reference that shows that this "fluke" is categorical (so not a fluke) - that is awesome. If you find it, please shot it to me! Thanks for sharing!

[@Jaylooker]

Hopf algebra diagrams when thickened look like topological quantum field theory (ie “pair of pants” and Y diagram). These diagrams with black and white dots look a lot like dessin d’’enfants. Doing some searching the connection looks like quasitriangular Hopf algebra (Drinfield) H H-colored ribbon connected bicolored ribbon graphs as Beyli pairs of dessin d’enfants (Grothendieck). See “Reimann surfaces, ribbon graphs and combinatorial classes” by Gabriele Mondello 2.2.4 and also note Miller in 5.3.2.

[@VisualMath]

Fantastic! Which one mimics which? I guess Hopf algebras were first?

[@pseudolullus]

​@@VisualMath Hopf algebra is actually used in topological computing proposals (by Kitaev, for instance), so it's all nicely connected, at least practically 😃

[@Jaylooker]

@@VisualMath Probably Hopf algebras. I also realized Hopf algebras relate back to motivic homotopy and chromatic homotopy I was discussing earlier with (mod p) Steenrod algebras, Morava K-theory found at each chromatic level, Eilenberg-MacLane spaces and their spectra. Maybe this could be used to derive Bloch-Kato conjecture in some form by building chromatic homotopy as before (Voevodsky) and showing each Morava K-theory at each level is a Hopf algebra and by extension a dessin d’enfant with action of absolute Galois group over the rationals.

[@VisualMath]

@pseudolullus and Jaylooker Thanks, both of you. I have worked with Hopf algebras all of my life, but I haven't seen the applications/connections you mention. Time to open the literature!

[@Jaylooker]

@@VisualMath No problem 👍

[@AkamiChannel]

Funny that I just searched "Hopf algebra" and found this video you made two weeks ago! You make good content. You should get a blue yeti mic or a lapel mic if you can afford it.

[@VisualMath]

Thanks for finding me ;-) And thank you for coming and the feedback - I will have a look into that.

[@AkamiChannel]

@@VisualMathI've watched your content many times, that's why it was funny to me. Btw, would love it if you could make videos on Bott periodicity. Another thing I've watched videos on many times, but never actually intuitively understood is why the next step up above the complex numbers has to be the 4-dim quaternions rather than something 3-dim. Yoroshiku!

[@VisualMath]

@@AkamiChannel Ah, food for thoughts! Yep, Bott periodicity or the uniqueness pof the tower R in C in H in O are nice topics. I will give those some thoughts. Thanks!

[@camerontorrance1992]

From my naive point of view the killer app of a coalgebra is that there natural algebra structure on linear maps C to A (C coalgebra and A algebra). Looking at it from this pov it is natural to ask if there is an inverse to the identity map viewed in the convolution algebra on Hom(H,H) for bialgebra H and if it exists it will satisfy the antipode axiom.

[@VisualMath]

Yes, that is a nice explanation. But I feel its still in hindsight - not sure whether I should believe from that point of view that there are trillions of Hopf algebras. But I - not you - have a naive point of view 🤨

[@camerontorrance1992]

@@VisualMath You can more or less get the group algebra and universal enveloping algebra examples by what hopf algebras generated by grouplike elements and primative elements respectively look like. The antipode identity is basically the inverse axiom for a group object in (Alg_k)^op with the main obstruction being that the antipode isn't algebra hom unless the hopf algebra commutative. These kinds of considerations will lend you think about about coordinate rings of algebraic groups (think there is a similar story with formal groups and cocommuative hopf algs but don't know the details).

[@VisualMath]

@@camerontorrance1992 That is great, thanks for explaining! But I still feel its in hindsight, sorry 😉 Hopf algebras came from cohomology rings, which makes a bit more sense to me because you get a bunch of example for free. Here the antipode is inverse.

In the last slide, in the Rels box, I guess there is a problem with the diagram in the center, because the LHS is from HHHH to HHHH, whereas the RHS is from HH to HH.

[@VisualMath]

Ah, good catch, thanks! Indeed, a second "dumbbell" next to the displayed one is missing. I have put a warning in the description.

[@peterwaksman9179]

Thanks

[@VisualMath]

You are welcome ☺

[@qschroed]

The diagrams at the theorem slide remind me a lot of string diagrams, is that intended?

[@VisualMath]

Good observation; that is indeed no coincidence: representations of Hopf algebras are prototypical examples of monoidal categories. The latter is the origin of string diagrams. In that sense Hopf algebras are prototypical examples of objects where string diagrammatics applies.

[@trongtue8384]

Hello, I it maybe not about your topic but can i asked you something ? Does mathematics need to do symbolic manipulation in their proof ?

[@VisualMath]

Hmm, that is bit of a meta question ("What is symbolic manipulation?, What is mathematics?"), so I can sadly only tell you my opinion instead of "the answer", sorry for that. I think, yes absolutely, mathematics is essentially equivalent to symbolic manipulation.

[@DB-nl9xw]

so interesting to watch and not able to understand anything, is it possible to make an easy way to understand?

[@VisualMath]

Sorry to hear that! I am not sure how this can be explained in an easier way (I guess some topics have prerequisites), but I would be happy to be convinced otherwise.