All Questions
Tagged with hopf-algebras category-theory
53
questions
4
votes
0
answers
56
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A diagrammatic proof of antipode being antihomomorphism in a Hopf algebra
Let $(H, \mu, \eta, \Delta, \epsilon, S)$ be a Hopf algebra with $S: H \to H$ denoting the antipode. By definition, $S$ is the convolution inverse of $1: H \to H$ in $\operatorname{End}(H)$, with the ...
- 1,350
1
vote
1
answer
38
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The 'union of factors' comultiplication in a monoid ring?
let $\mathbb{Z}[M]$ be 'the' monoid ring of $M$ over $\mathbb{Z}$; that is to say (if my understanding is right) the set of finite linear combinations of elements of $M$, with product given by $\sum_n ...
- 52.4k
1
vote
1
answer
90
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Understanding a proof of a lemma for rigid categories [closed]
I'm reading the proof of the lemma 3.4 in the Bruguieres' paper on Hopf monads which claims the following:
Lemma Let $F,G: \mathcal{C} \rightarrow \mathcal{D}$ be two strong monoidal functors and $\...
4
votes
0
answers
38
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Where does the unitarity structure of $U_q(\mathfrak{sl}_2)$ come from?
It is known that when $q$ is the root of unity, the representation of the quantum group $U_q(\mathfrak{sl}_2)$ is a unitary modular tensor category. However, if we want it to have the dagger structure,...
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3
votes
1
answer
106
views
Cocommutative bimonads: Why does this diagram commute?
1. Definitions
Let $(C, \otimes,I, a, l,r,c)$ be a monoidal category with braiding $c:\otimes \rightarrow\otimes ^{op}$. Let $(S,\mu,\eta,\tau,\theta)$ be a bimonad on $C$.
Following Turaev and ...
- 3,068
3
votes
0
answers
118
views
Proof of Tannaka recognition theorem
I am trying to prove the theorem 5.12.7 in the book “Tensor categories” by Etingof, Gelaki, Nikshych and Ostrik.
The statement is as follows:
The assignments
$(\mathcal{C},F) \mapsto H = \mathrm{End}(...
- 31
7
votes
1
answer
249
views
Yoneda's lemma: group morphisms give Hopf-algebra morphisms
Let $k$ be a commutative ring. Let $\text{Alg}$ be the category of commutative $k$-algebras and $\text{CHopf}$ the category of commutative Hopf-algebras. Let us also write $[\text{Alg}, \text{Grp}]$ ...
user839372
1
vote
1
answer
100
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Comultiplication arising as pullback for group representations
Let $V$ and $W$ be $k[G]$-modules for an algebraically closed field, $k$ and $G$ a group. Now I know that $V\otimes W$ has a $k[G]\otimes k[G]$-module structure, and because $k[G]$ is in fact a Hopf ...
- 1,776
1
vote
0
answers
98
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What are other examples of pivotal non-spherical categories?
Apparently, an example of a pivotal non-spherical category can be found in the world of Hopf algebras/representation theory:
Let $(H, \omega)$ be a pivotal non-spherical Hopf algebra with pivot $\...
- 3,068
5
votes
0
answers
181
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Epimorphisms and monomorphisms in the categories of Hopf algebras
From this paper I learned that in the category $\operatorname{HopfAlg}$ of Hopf algebras over a field $k$ epimorphisms are not necessary surjective and monomorphisms are not necessary injective. Can ...
- 2,463
2
votes
1
answer
101
views
Categorization of Group Scheme of rank 2
I got stuck on this Exercise 2.11 in the book Introduction to Affine Group Scheme of Waterhouse. I really appreciate if anyone can give me a hint.
Let $A$ be a Hopf algebra over $k$ (a base ring) ...
- 4,845
2
votes
1
answer
130
views
Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$ mod $I \otimes I $ for all $x \in I$.
I got stuck on this problem, so if anyone can give me a hint on this, I really appreciate.
Let $I$ be the augmentation ideal in Hopf algebra $A$. Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$
mod $...
- 4,845
1
vote
0
answers
91
views
End of a category
I am trying to prove Proposition 7.2 from this paper:
$\textbf{Proposition 7.2:}$ Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $\mathbb{K}$. The end object $\Gamma$ in $\textbf{Rep ...
- 2,391
3
votes
1
answer
150
views
Is there a consistent way to get all possible coproducts?
Let's illustrate the problem with an example. Consider an algebra of polynomials in one variable $1,x,x^2,\ldots$ with the product $\nabla (x^i,x^j) = x^{i+j}$. Then, reversing arrows in the diagram
$\...
- 403
4
votes
0
answers
93
views
String diagram of left $H$-action on $\mathrm{Hom}(U,V)$
1. Context
Let $\mathbb k$ be a field. Let $H$ be a $\mathbb{k}$-Hopf algebra. Let $U, V$ be objects in the category $H\text{-}\mathrm{mod}$ of left $H$-modules. (In particular they are $\mathbb{k}$-...
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