Timeline for Why do Groups and Abelian Groups feel so different?
Current License: CC BY-SA 3.0
9 events
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| Jun 11, 2016 at 22:21 | comment | added | Todd Trimble | Do those books really say that, and not that it's an antihomomorphism? Actually you do not need to assume that, only that it's a linear map which obeys the usual antipode equation, and it turns out that being an antihomomorphism (at both the algebra and coalgebra level) is a consequence. A derivation of this fact is given here: ncatlab.org/nlab/show/Hopf+algebra#definition Anyway, I think your surmise (L.Z.) is correct. | |
| Jun 11, 2016 at 21:03 | history | edited | L.Z. Wong | CC BY-SA 3.0 |
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| Jun 11, 2016 at 21:02 | comment | added | L.Z. Wong | Thanks for the clarication! I've struck out the erroneous parts of my answer. Is it still true that a Hopf algebra is commutative iff it's antipode is an algebra homomorphism? I've come across books where the antipode is defined to be an algebra homomorphism - surely that can only hold for commutative Hopf algebras. | |
| Jun 11, 2016 at 20:53 | history | edited | L.Z. Wong | CC BY-SA 3.0 |
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| Jun 11, 2016 at 20:48 | comment | added | Qfwfq | Oh wait, I guess I wanted to say what Todd Trimble said in the last sentence and I even messed up | |
| Jun 11, 2016 at 20:47 | comment | added | Qfwfq | Isn't a Hopf algebra a group object in $(k-\mathbf{Mod})^{\mathrm{op}}$ (as opposed to $k-\mathbf{Mod}$)? | |
| Jun 11, 2016 at 20:38 | comment | added | Todd Trimble | Group objects make sense in cartesian monoidal categories, i.e., monoidal categories where the tensor product is a cartesian product. The point is that you need projections and diagonal maps to internalize the group axioms. For instance, the category of cocommutative coalgebras over $k$ has cartesian product given by $\otimes_k$. But $\otimes_k$ for $k$-Mod doesn't work like that. So I think one should rather say that cocommutative Hopf algebras are group objects in cocommutative coalgebras, and commutative Hopf algebras are cogroups in commutative $k$-algebras. | |
| Jun 11, 2016 at 19:00 | history | edited | L.Z. Wong | CC BY-SA 3.0 |
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| Jun 11, 2016 at 18:49 | history | answered | L.Z. Wong | CC BY-SA 3.0 |