# Tagged Questions

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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### The Irreducible Representations of the Sekine Quantum Groups

A one parameter family of finite quantum groups are introduced by Sekine here. I am looking for a reference where the (co)representation theory is discussed. Such a reference may not exist --- Google ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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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### $q$-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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### 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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### 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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### '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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### Reference request: Lusztig's symmetries

Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$, \begin{align} ...
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### Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...
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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 ...
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### Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$

I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167: Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra ...
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 ...
In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of ...