Timeline for An algebra map between Hopf algebras that does not commute with the counit

5 events

when toggle format	what		by	license	comment
May 11, 2023 at 7:33	comment	added	Tom De Medts		@LorenzoDelVecchiopontopolos That depends on the field. For instance, the specific example I gave is indeed isomorphic to the group algebra $k\mathbb{Z}_2$ if $\operatorname{char}(k) \neq 2$, but not if $\operatorname{char}(k) = 2$. In general, the group algebra $k\mathbb{Z}_n$ is isomorphic to the Hopf algebra of functions from $\mathbb{Z}_n$ to $k$ if and only if the polynomial $f(x) = x^n - 1$ has $n$ different roots in $k$ (in other words, if it splits over $k$ and has no multiple roots).
May 10, 2023 at 20:17	comment	added	Lorenzo Del Vecchiopontopolos		Thanks for the reference. But I thought that for prime order, all finite-dimensional Hopf algebras were isomorphic . . .
May 10, 2023 at 18:24	comment	added	Tom De Medts		@LorenzoDelVecchiopontopolos It's not isomorphic to the group Hopf algebra of $\mathbb{Z}_2$. It's the Hopf algebra of functions from $\mathbb{Z}_2$ to $k$. (This is the second example on en.wikipedia.org/wiki/Hopf_algebra#Examples.)
May 10, 2023 at 17:03	comment	added	Lorenzo Del Vecchiopontopolos		Sorry but I don't see how this is isomorphic to the group Hopf algebra of $\mathbb{Z}_2$.
May 10, 2023 at 14:35	history	answered	Tom De Medts	CC BY-SA 4.0