All Questions
Tagged with qa.quantum-algebra monoidal-categories
14 questions with no upvoted or accepted answers
7
votes
0
answers
300
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Does the pentagon axiom force the associativity constraint to be a natural isomorphism?
Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...
7
votes
0
answers
172
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When is Rep(U_q(g)) invariant under q -> -q and why?
Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...
- 28.1k
4
votes
0
answers
320
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The fusion categories with a strict skeleton
We refer to the book Tensor Categories (by Etingof-Gelaki-Nikshych-Ostrik) for all the notions mentioned in this post.
A fusion category is skeletal if two isomorphic objects are always equal. Every ...
4
votes
0
answers
103
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Scaling Yetter--Drinfeld Modules
A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding ...
- 835
4
votes
0
answers
310
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Nichols Algebras as Braided Hopf Algebras
Given a Hopf algebra $H$ and a Yetter--Drinfeld module $V$ over $H$, it is well-known that $V$ has an induced braided vector space structure, and so, one can consider it's Nichols algebra which is a ...
- 159
4
votes
0
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89
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Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?
This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...
- 5,565
4
votes
0
answers
134
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Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras
As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...
- 1,453
4
votes
0
answers
300
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Reshetikhin-Turaev and links with a distinguished component
Hi,
This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question.
Let $T$ be the category whose objects are ...
- 8,524
3
votes
0
answers
99
views
Isomorphic objects have the same dimension (pivotal categories)
I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,
$$
\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...
- 143
3
votes
0
answers
109
views
Are the fusion categories weakly Frobenius?
A well-known open problem (generalizing Kaplansky 6th conjecture) asks whether every (spherical) fusion category $\mathcal{C}$ (over $\mathbb{C}$) is of Frobenius type, i.e. for every simple object $X$...
2
votes
0
answers
103
views
Questions about proof that all indecomposable module categories over $\operatorname{Rep}(G)$ are equivalent to $\operatorname{Rep}^1(H,\omega)$
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\End{End}$In Ostrik - Module categories, weak Hopf algebras and modular invariants, it is ...
- 301
2
votes
0
answers
178
views
Categorical dimension and formal codegrees
Let $\mathcal{C}$ be a complex fusion category. If it admits a pivotal structure $a$ then by [1, Proposition 4.7.12], $\dim_a$ induces a character $\chi$ on the Grothendieck ring $Gr(\mathcal{C})$, of ...
1
vote
0
answers
118
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Are there non-semisimple complex "non-unital special Frobenius algebras"?
I'm interested in "non-unital special Frobenius algebras", consisting of two linear maps (morphisms in the symmetric monoidal category of finite-dimensional complex vector spaces)
$$\mu: V\...
- 3,001
1
vote
0
answers
96
views
Frobenius Monoids as Collapsed 2-Categories
Let $\mathbf{COB}_2$ denote the 2-category given by
$\bullet$ objects are finite sets of points
$\bullet$ 1-morphisms between these are 1d cobordisms
$\bullet$ 2-morphisms are 2d cobordisms with ...
- 433