Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are ...

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Classifying Low Dimensional Solutions of the Yang--Baxter Equation

What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions? To make my question more specific, have all solutions for dimension $2$ and $3$ been ...
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56 views

Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by ...
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Cosemi-simple FRT Hopf Algebras

This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...
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134 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 ...
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How to compute $t_0$ and $r^0$ in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations?

I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations. In the book a guide to quantum groups, on page 83, there is an example of solutions of the ...
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86 views

Questions about Sklyanin bracket

For every classical r-matrix $r$, there is a Poisson bracket called Sklyanin bracket associated to $r$. It is defined in (3.3) of page 5 in (http://arxiv.org/pdf/1101.0015v2.pdf) as follows. ...
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57 views

Relation between quantum affine algebras and W-algebras

In the paper, there is a concept $qq$-characters for W-algebras $W_{q_1,q_2}$. The usual $q$-character is defined for quantum affine algebra in the paper. A $q$-character can be obtained from a ...
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Why are quantum groups so called?

I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
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153 views

Brauer-Picard for a fusion category coming from a quantum group

In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A ...
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166 views

$U_q(\mathfrak{sl}_2)$ representations of “quantum dimension” zero

I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U_q(\mathfrak{sl}_2)$ and I ...
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73 views

What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$?

What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$ in terms of $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$? Any help will be greatly appreciated!
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Center of $U_q(sl_3)$ and $U_q(sl_4)$

In the book a guide to quantum groups, page 285, the center of $U_q(g)$ is described in Theorem 9.1.6. The center of $U_q(sl_2)$ is computed explicitly in Example 9.1.7. I tried to compute the center ...
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87 views

Number of cluster variables

In the paper cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}_2$ for ...
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58 views

Homomorphisms of quantum spaces

Suppose we have a look at the Hopf $*$-algebra $U_q(sl(2))$ and the Hopf $*$-algebra $A_q(\widetilde{S}^3)$ introduced in the paper: http://arxiv.org/pdf/q-alg/9605017v1.pdf. I want to find a relation ...
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94 views

Compact Quantum Groups and FRT-Algebras

As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which ...
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128 views

Representations of the algebra of odd quantum spheres

I read the article by Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf). Here they describe in section $4$ what the algebra of functions on the total space of a family of quantum ...
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68 views

Irreducible representations of quantum affine algebras

The finite dimensional simple modules of quantum affine algebras are parameterized by Drinfeld polynomials. Some other modules of quantum affine algebras are studied in the paper. Some infinite ...
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169 views

Center of quantum affine algebras

Are there some references about the center of quantum affine algebras? I searched on google and only find the paper. In particular, what is the center of $U_q(\widehat{\mathfrak{sl}_2})$. Thank you ...
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1answer
212 views

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 ...
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115 views

An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra. There already exists a generalization of Cauchy theorem using exponent, see here. We are interesting in an alternative ...
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Verifying Woronowicz’s proof that all $ {\text{SU}_{q}}(2) $’s are isomorphic as $ C^{*} $-algebras, where $ -1 < q < 1 $

This question is related to one that I asked some time ago. Definition 1. Let $ q \in (-1,1) $. Let $ A $ be a unital $ C^{*} $-algebra and $ (x,y) $ a pair of elements of $ A $. Then define the ...
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136 views

What are the applications of the depth 2 reduction to the subfactors theory?

Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth $2$ ...
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1answer
89 views

Why $A^*A =A$ implies that $A$ is a C$^*$ algebra (Proposition 5.2.8 of An Invitation to Quantum Groups and Duality by Thomas Timmermann)

I am reading "An Invitation to Quantum Groups and Duality From Hopf Algebras to Multiplicative Unitaries and Beyond" by Thomas Timmermann. In the proposition 5.2.8 (page 117) the author provide a ...
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36 views

Deforming the category of representations of the Yangian of a simple Lie algebra?

Since I got a very good answer to my previous question, 217585, I am asking a sequel by moving up a level. Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and ...
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98 views

Deforming the category of representations of a simple Lie algebra?

This is a follow up question to the insightful answer by Theo Johnson-Freyd to the question 105221. This answer explained that the quantised enveloping algebra, $U_q(\mathfrak{g})$, (defined by a ...
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105 views

What's the relation between half-twists and star structures on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
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388 views

Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?

The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...
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1answer
58 views

Radicals of co-quasitriangular map

Let $B=\mathcal{O}_R\left(GL(n)\right)$ be a localization of the algebra $A(R)$ of functions on the quantum formal group corresponding to the matrix $R$ ["Quantization of Lie groups and Lie algebras", ...
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45 views

Co-quasitriangular Hopf algebra - notation

In one article I found the following statement : If $A$ is a Hopf algebra over $k$ with co-quasitriangular structure $r$, then one can define map $r_{21}*r:A\otimes A\rightarrow k \ \ $ (...). ...
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Symmetries of module categories over the category of representations of quantum $sl(2)$

The category $\mathcal{C}_l$ of tilting modules of the quantum group $U_q(sl_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when ...
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290 views

A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $

Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...
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Is there an E8 symmetry in the zero-field Ising model?

In the paper On classification of modular tensor categories by Rowell, Stong and Wang, they list the Ising modular category $I$ as having 3 objects $1$, $\sigma$ and $\psi$, with fusion rules ...
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1answer
97 views

'Test Functions' to Lower Bound the Norm of Elements of Dual Quantum Group

There may well be an answer to this question in a simpler category than that of finite dimensional quantum groups and in that case this question is more suitable to math.stack and I apologise in ...
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131 views

Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) ...
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59 views

quantum deformation

The standard quantum groups, say $GL_q(n)$ or $U_q(gl(n))$, depend on the parameter $q$, which in the classical limit tends toward 1. Let $t:=q-1$ be considered as a generic parameter then we can ...
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Intuition behind the definition of quantum groups

Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to ...
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109 views

Help finding paper: De Concini, Kac - Quantum Groups at roots of 1

I am looking for a specific paper, that I have found very difficult to trace. C. De Concini, V. Kac - Quantum Groups at roots of 1 Specifically, the paper is cited as follows (on De Concini's ...
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1answer
137 views

How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module?

Context Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the ...
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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? ...
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1answer
498 views

What is the current state of generalizations Noether's theorem?

The well-known Noether's theorem is a vital tool in classical physics. But it assumes some hypothesis, many of which could be removed by a detailed look. So my question is: In what directions has ...
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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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507 views

exceptional cases in Kazhdan-Lusztig

The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine). What's special about those cases?
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204 views

The coproducts $\mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ and $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$

A coproduct $\varphi: \mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ is given by: $x \mapsto 1 \otimes x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}_q[U]$. There is a ...
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In Algebraic Compact Quantum Groups, is an Irreducible Corepresentation equivalent to its Conjugate?

A quantum group $A$ here is an algebraic compact quantum group --- a Hopf*-algebra with a Haar State. Here $\hat{A}$ is the set of linear functionals $\{\mathcal{F(a)}:a\in A\}$ of the form ...
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Clebsch Gordan coefficients of compact quantum groups

Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
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172 views

Deformation quantization of a closed Riemann surface with genus >1

Quantization of of an elliptic curve can be done in different ways. In C^*-algebraic version, one can start with the C^*-algebra ...
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195 views

Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
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579 views

$q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?

Define n-quantum vector space to be the algebra $$ {\mathbb C}_q^n := \mathbb{C}\left< x_i \mid i =1, \ldots, N\right>/\left<x_i x_j = q x_j x_i \mid i<j\right>. $$ For $q=1$, we get ...
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1answer
125 views

Comodule analogue for statement that a faithful representation of an affine group scheme generates all

If $V$ is a faithful finite dimensional representation of an affine group scheme $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n ...
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The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual. Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...