All Questions
21 questions
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What is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right] _{\varepsilon _i}$ in the restricted specialization in QUE algebras?
I have a question about the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. It comes from section $9.3$ on page $300$ of this book
In Section 9.1, the authors define ...
- 1,846
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0
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267
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Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
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0
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207
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parameter of a quantum group
I am currently learning about quantum groups, and I got a question about how two different ways of thinking the $q$-parameter of quantum groups are related to each other. Here by a quantum group, I ...
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0
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Noncommutative group schemes corresponding to quantum groups
I'm not an expert on quantum groups by any stretch, so forgive me if this question seems overly naive. That said, I was wondering if there is a way (or if there has been any attempt in the literature) ...
- 1,516
6
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1
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Reference request : table of quantum Clebsch-Gordan coefficient
From a quick Google search, one can find a table of the first Clebsch-Gordan coefficient. For example this table. Those are used to pass between the tensor product bases and the bases as sum of ...
- 188k
6
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1
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172
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Norm of contragredient of unitary representations of compact quantum groups
Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms.
Let $G = (A, \Delta)$ be a ...
- 4,351
6
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1
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226
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Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
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752
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Where can I find Drinfeld's original papers on quantum groups?
Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $U_h(\mathfrak{g})$ be the Drinfeld-Jimbo quantum group, i.e. the $\mathbb{C}[[h]]$-algebra topologically generated by $X_i,Y_i,H_i$ where $1\leq i\...
- 1,066
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3
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442
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Commutative and Cocommutative Quantum Groups
I am using this definition:
An algebra of functions on a finite quantum group $\mathbb{G}$ is a finite dimensional $C^\ast$-Hopf algebra $A=:F(\mathbb{G})$.
I have the following (very well known --...
- 2,121
7
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2
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405
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The Irreducible Representations of the Sekine Quantum Groups
Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
$$\mathcal{B}_n=\mathbb{Z}_n\times\mathbb{Z}_n=\{(i,...
CommunityBot
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0
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Representations of quantum groups at roots of unity
I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
CommunityBot
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11
votes
2
answers
1k
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Yang–Baxter explanation
What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? ...
- 3,078
1
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0
answers
115
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Are the Standard Quantum Groups Coordinate Rings Noetherian?
Are the standard quantum groups $C_q[G]$ Noetherian and if so what is a standard reference?
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7
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1
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672
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The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
I asked this question on Math.Stack but have not had any answers.
Question
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
The trivial ...
CommunityBot
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8
votes
1
answer
358
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Abel's five terms relation from Yang-Baxter equation?
Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations?
If yes, how?
Thanks for any help.
- 24.9k
7
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1
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321
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Real forms of Drinfeld-Jimbo quantum groups
A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and ...
- 8,559
13
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7
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Open problems in the theory of compact quantum groups
What are the important open problems in the theory of compact quantum groups? Or conjectures?
Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules ...
CommunityBot
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24
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9
answers
3k
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expository papers related to quantum groups
Hello all,
I know basic representation theory(finite groups, lie groups&lie algebras) and I want to get a flavor of quantum groups (why they are useful, important results etc) and other related ...
- 52.9k
12
votes
1
answer
834
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R-matrices, crystal bases, and the limit as q -> 1
I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of ...
- 8,559
3
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1
answer
344
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Reference for the Hecke relation for the universal R-matrix
I've come across a reference in a paper to the
Hecke relation for the universal R-matrix of a quasi-triangular Hopf algebra.
I've looked around, standard references, online etc, but can't seem to ...
- 1,568
7
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2
answers
1k
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Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules
Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 ...
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