All Questions
Tagged with quantum-groups ct.category-theory
29 questions
0
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0
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44
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Categorical duals for Yetter-Drinfeld modules [duplicate]
Yetter-Drinfeld (YD) modules appear naturally in the theory of Hopf algebras. They are both modules and comodules at the same time, satisfying a certain compatibility condition, as presented here. The ...
4
votes
0
answers
68
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Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent ...
- 727
3
votes
2
answers
113
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Does unitarity and modularity constrain fusion multiplicities to be 0,1?
If I have a braided tensor category that's unitary and modular, then how does the unitarity and modularity constrain the fusion multiplicities?
I know that if $a,b,c \in ob({C})$ satisfy the fusion ...
- 569
1
vote
0
answers
121
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Lagrangian subcategories of (non-pointed) braided tensor categories
I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...
- 113
3
votes
1
answer
178
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Adding finite direct sums to a C*-tensor category
Consider the following fragment from the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset (p72, in section 2.5):
$\ \ \ $ Assume $\mathscr{C}$ is a ...
- 175
7
votes
1
answer
315
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Does the category of commutative and cocommutative Hopf algebras have enough injectives?
It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see https://arxiv.org/abs/1502.04001v2 and its references). But does it have enough injectives? What about ...
- 3,079
5
votes
1
answer
148
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Exactness of functors in a $C^*$-tensor category
I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. In section 2.3 "Fiber functors and reconstruction theorems*, the following ...
- 175
2
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0
answers
514
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Relation Hopf categories and categorified quantum groups
In the paper Hopf Categories Crane and Frenkel gave a definition of a Hopf category, which they considered as a categorification of a quantum group. Categorifications of quantum groups have later been ...
- 183
3
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0
answers
119
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Is the category of Yetter-Drinfeld modules abelian?
Is $YD(H)$ the category of Yetter--Drinfeld modules over a Hopf algebra (defined over a field $k$) necessarily abelian? If not then what is the simplest example of a Hopf algebra $H$ for which $YD(H)$ ...
- 1,144
4
votes
0
answers
208
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Are the finite quantum permutation groups, weakly group-theoretical?
Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...
2
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0
answers
165
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Frobenius reciprocities
An adjunction of the form $\mathrm{Hom}(A \otimes X, Y) \cong \mathrm{Hom}(X, A^* \otimes Y)$ in a rigid monoidal category is sometimes called Frobenius reciprocity. Is there a result that unifies ...
- 974
7
votes
2
answers
631
views
Abelian category from the category of Hopf algebras
The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf
sub-algebra of $H_1$. Is there some replacement or alteration of the notion
of a kernel in the Hopf algebra setting. Same ...
- 1,144
3
votes
1
answer
241
views
F-symbols for compact Lie groups
Consider a Unitary Modular Tensor Category constructed from the quantum group of some compact, simple and simply-connected Lie group $U_q(G)$ at $q=e^{2\pi i/(k+h)}$ for some integer $k$. In general, ...
- 195
6
votes
0
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338
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Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?
Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...
- 63.9k
10
votes
3
answers
856
views
Tannaka-Krein duality in Chari-Pressley's book
I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here.
V.Chari and A.N.Pressley in their "Guide to Quantum ...
- 7,426
26
votes
1
answer
2k
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Have the Quantum Group Theorists taught the Group Theorists Anything?
I will start with the general before moving to the specific.
Consider for a moment the two (very) soft definitions.
An abstraction of an object $X$ is a category $\mathcal{C}_0$ such that $X$ ...
- 1,037
11
votes
3
answers
663
views
Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules
In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra:
The space of ...
- 219
5
votes
1
answer
498
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Quantum groups at $q=-1$
For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...
- 219
9
votes
2
answers
362
views
Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups
For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
- 219
7
votes
1
answer
556
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Quadratic algebras, quadratic operads, quadratic categories and quantum cohomology
Motivated by the quantisation of the symmetric laws in physics, the category of quadratic algebras has been endowed with two tensor products by Manin in his Montreal lectures notes. These products ...
- 3,652
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
5
votes
1
answer
347
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Category of bicomodules of a cosemisimple Hopf algebra
A cosemisimple Hopf algebra $H$ is one which is equal to the direct sum of its subcoalgebras. As is well-known, this is equivalent to its category of $H$-comodules being semisimple. Is this also true ...
- 459
4
votes
1
answer
347
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Fusion Rules for Quantum Groups
For the Drinfeld--Jimbo quantum groups $U_q(\frak{g})$, we have an equivalence of categories between the representations of $U_q(\frak{g})$ and the representations of $U(\frak{g})$.
Is this a ...
- 1,479
9
votes
2
answers
1k
views
Algebra in a category
I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \...
- 6,201
7
votes
2
answers
623
views
How to make a premodular category a modular tensor category?
A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the ...
- 875
18
votes
0
answers
577
views
Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?
Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...
- 118k
4
votes
3
answers
966
views
When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?
Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...
- 5,565
3
votes
0
answers
177
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quantum deformations of tensor category
I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...
- 1,457
8
votes
5
answers
1k
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Braided Monoidal 2-categories with duals
Which categorifications give explicit braided monoidal 2-categories with duals?
This question is in response to Ben Webster's questions in recent history. The point is that given a braided monoidal 2-...
- 5,264