Group-like elements and antipode

Stream: maths

[Yaël Dillies]

In Toric, we need results about group-like elements of a Hopf algebra. If $H$ is a Hopf algebra, then $a ∈ H$ is group-like if

• counit a = 1
• comul a = a ⊗ₜ[R] a

[Yaël Dillies]

(thanks to @Julian Külshammer for pointing out that my original definition was wrong for coalgebras/bialgebras!)

[Yaël Dillies]

I have proved the following theorems:

• If a is group-like, then antipode R a is group-like too
• If a is group-like, then antipode R (antipode R a) = a

[Yaël Dillies]

This makes it feel like the first theorem should actually have a converse:

• If antipode R a is group-like, then a is group-like too

[Yaël Dillies]

But of course from antipode R a being group-like and the second theorem, we get that antipode R (antipode R (antipode R a)) = antipode R a, while one would instead want antipode R (antipode R a) = a. Since the antipode needn't be injective in general, I am starting to suspect that my conjectured converse is false.

[Yaël Dillies]

Can anyone come up with a counterexample?

[Andrew Yang]

(I think there’s a typo in “while one would instead want”)

[Kevin Buzzard]

Antipode isn't injective in general? Isn't this inverse on the group side of the story? (I should say I'm only familiar with commutative Hopf algebras)

[Yaël Dillies]

Indeed, this fact is surprising but it is a fact: https://mathoverflow.net/questions/417501/hopf-algebra-with-a-non-invertible-antipode

[Edward van de Meent]

Yaël Dillies said:

But of course from antipode R a being group-like and the second theorem, we get that antipode R (antipode R (antipode R a)) = a, while one would instead want antipode R (antipode R a) = antipode R a. Since the antipode needn't be injective in general, I am starting to suspect that my conjectured converse is false.

are you missing an antipode on the rhs here? (3 left, 1 right)?

[Yaël Dillies]

Sorry yes, I edited the wrong RHS after Andrew pointed out the typo :woman_facepalming:

[Yaël Dillies]

Kevin Buzzard said:

Antipode isn't injective in general? Isn't this inverse on the group side of the story? (I should say I'm only familiar with commutative Hopf algebras)

Wikipedia says that a Hopf algebra is involutive (ie has an involutive antipode) whenever it is either commutative or cocommutative. So your intuition is safe

[Kalle Kytölä]

One place with a short proof of this (commutativity or cocommutativity implies involutivity of the antipode) using convolution algebras (which I hope are a tool that @Yaël Dillies finds natural given that they formalized Möbius inversion) is my Hopf algebra lecture notes, Corollary 3.41 (sorry about the self-plug).

(These notes also contain a counterexample to involutivity and a characterization of involutivity left as an exercise. :grinning: There are also a fair amount of properties of grouplike elements if I recall correctly from my younger self's writings. :older_man:)

[Markus Himmel]

mathlib has this for Hopf monoids in monoidal categories: docs#Hopf_Class.antipode_antipode The proof looks very similar to your Corollary 3.41.

[Kevin Buzzard]

But you only care about commutative Hopf algebras for Toric, right?

[Yaël Dillies]

Yes, correct, this is just musing about the correct API mathlib should have, and irrelevant to Toric