# Convolution in Hopf algebras

For each Hopf algebra $H$ its space ${\mathcal L}(H)$ of operators $A:H\to H$ is usually endowed with the operation of convolution by the identity $$A*B = \mu \circ (A\otimes B)\circ \varDelta$$ where $\mu$ is the multiplication, $\varDelta$ the comultiplication in $H$, and $A,B\in{\mathcal L}(H)$.

Can anybody advise me a text where this notion is described in detail? I wonder, in particular, which identities connect $A*B$ with the usual operation $A\circ B$ of composition of operators.

• Sergei, i have added i couple of tags. But if you disagree feel free to retag. Commented May 9, 2018 at 2:02
• No, it's OK! I agree. Commented May 9, 2018 at 4:06

## 1 Answer

You can try "Hopf algebras. An introduction" by Dascalescu, Nastasescu, Raianu. They have a detailed account with full proofs and further examples and exercises.

Kassel's book "Quantum groups" and Sweedler's book on "Hopf algebras" (see esp. p. 69-79, 314-315) might also prove useful.

Finally, you might find some interest at https://math.stackexchange.com/a/1980032/195021.

Edit: Motivated by your question on the relation between convolution and the composition, i could not find some direct reference, however i recalled some old notes of mine and found the following relations (in what follows, $\mu$, $\Delta$, $\eta$, $\varepsilon$, $S$, stand for the corresponding multiplication, comultiplication, unity, counity and antipode maps respectively):

1. the composition distributes over convolution: i.e. if $f,g\in Hom(F,G)$, $p\in Hom(H,F)$ where $H,F,G$ are Hopf algebras and $Hom(.,.)$ stands for the corresponding Hopf algebra maps, then: $$\boxed{ (f\star g)\circ p=(f\circ p)\star(g\circ p) }$$ Proof: by definition: $$(f\star g)\circ p=\mu_G\circ(f\otimes g)\circ\Delta_F \circ p$$ and $$(f\circ p)\star(g\circ p)=\mu_G\circ\big((f\circ p)\otimes (g\circ p)\big)\circ\Delta_H = \mu_G\circ (f\otimes g)\circ (p\otimes p)\circ\Delta_H =\\=\mu_G\circ(f\otimes g)\circ\Delta_F\circ p$$ where in the last line we have made use of the fact that since $f,g,p$ are assumed to be Hopf algebra morphisms they are also coalgebra morphisms and thus $\Delta_F \circ p=(p\otimes p)\circ\Delta_H$.
2. $f\circ S_F=S_G\circ f$, is the $\star$-inverse of $f$, i.e.: $$\boxed{ (S_G\circ f)\star f=f\star(S_G\circ f)=\eta_G\circ\varepsilon_F}$$ Proof: Applying the definition of the convolution we get: $$\big((S_G\circ f)\star f\big)=\mu_G\circ\big((S_G\circ f)\otimes f\big)\circ\Delta_F= \mu_G\circ(S_G\otimes Id)\circ(f\otimes f)\circ\Delta_F=\\=\mu_G\circ(S_G\otimes Id)\circ\Delta_G\circ f=\eta_G\circ\varepsilon_G\circ f=\eta_G\circ\varepsilon_F$$ where we have used:
$\bullet \ (f\otimes f)\circ\Delta_F=\Delta_G\circ f$, because $f\in Hom(F,G)$,
$\bullet \ \mu_G\circ(S_G\otimes Id)\circ\Delta_G=\eta_G\circ\varepsilon_G$, by the definition of the antipode,
$\bullet \ \varepsilon_G\circ f=\varepsilon_F$, because $f\in Hom(F,G)$.

3. A more general property, which has a categorical "flavor" and is related to the preceding properties, is the following one:

Let $\mathcal{H}$ be the Category of commutative, cocommutative, finite dimensional Hopf algebras. Let the corresponding Hom-sets $\mathcal{H}om(F,G)$ be equipped with sum (convolution) and product (composition). Then, $\mathcal{H}$ becomes an abelian Category.
(see Sweedler's book, sect. 16.2, p. 314-315, where this is result is cited).

What is essentialy happening here, is that $\mathcal{H}om(F,G)$ becomes an abelian group -under convolution- with neutral element: $\eta_G\circ\varepsilon_F$.
Furthermore, it can be shown that there is an equivalence of abelian Categories, between the Category of commutative, cocommutative, finite dimensional Hopf algebras $\mathcal{H}$ and the category $\mathcal{Ab}_{fin}$ of the finite, abelian groups. It is possible to construct a fully faithful and essentially faithful functor between $\mathcal{H}$ and $\mathcal{Ab}_{fin}$. See more details on this point at: Reference on correspondence between commutative Hopf Algebras and Groups

• As far as I understand, Dascalescu, Nastasescu and Kassel prove this only for the Hopf algebras in the category of vector spaces, since they use the Sweedler notations. I thought there must be a general proof for all symmetric monoidal categories. Also are there any identities that connect $A*B$ with $A\circ B$? Commented Apr 28, 2018 at 6:05
• @Sergei, i have updated the post including some more detailed info. Hope, it is more helpful now. Commented Apr 29, 2018 at 16:11
• KonKan, that is interesting. Several notes: 1) in the first identity $f$ and $g$ seem to not be homomorphisms of Hopf algebras, just morphisms in the given category, 2) I don't understand how you prove the second identity, 3) this needs to be published. Commented Apr 30, 2018 at 6:43
• Sergei, I have added the proof of the second identity and some more details in 3). Commented May 1, 2018 at 15:11
• KonKan, thank you! You should publish all this in a book. Commented May 1, 2018 at 16:19