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7 votes
2 answers
201 views

Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra always have finite order?

Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let $B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
13 votes
1 answer
281 views

Finiteness of the number of Hopf subalgebras

Let $H$ be a finite-dimensional Hopf algebra over the complex field. Question: Does $ H $ have a finite number of Hopf subalgebras? In the case where $ H $ is semisimple, the answer is yes. According ...
1 vote
1 answer
64 views

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 ...
2 votes
1 answer
78 views

Does there exist a nontrivial triangular weak Hopf algebra?

Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair ($H,\mathcal{R}$) where $H$ is a WHA and $\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(...
3 votes
0 answers
267 views

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 &...
6 votes
0 answers
349 views

Quantum Hilbert's fifth problem

Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case. The definition of a quantum ...
4 votes
0 answers
82 views

Is the Drinfeld element of a semisimple quasitriangular Hopf algebra invariant under the Drinfeld twist?

Let $A$ be a finite dimensional semisimple quasitriangular Hopf algebra over $\mathbb{C}$ with universal $R$-matrix denoted by $\mathcal{R}\in A\otimes A$. The Drinfeld element of $A$ is defined as $$...
4 votes
0 answers
183 views

Does a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?

Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations: \begin{align*} & ac = e^{-h}ca, \quad bd = e^{-h}...
2 votes
0 answers
132 views

A question about q-binomials at roots of unity

I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization&...
1 vote
0 answers
54 views

Coproduct on $U_q(sl_2)$

Recall that $U_q(sl_2)$ is the quotient of the free associative $\mathbb{Q}(q)$-algebra on generators $E$, $F$, $K^{\pm 1}$ such that $KE = q^2 EK$, $KF = q^{-2} FK$, and $[E,F] = \frac{K - K^{-1}}{q -...
3 votes
0 answers
74 views

Are all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact quantum groups?

Let us consider the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$. Under what assumptions about $\mathfrak{g}$, does the enveloping algebra generate a locally ...
4 votes
0 answers
68 views

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 ...
7 votes
3 answers
650 views

Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
1 vote
0 answers
70 views

Affiliating the whole algebra of 'coordinates' with a locally compact quantum group

When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$) ...
6 votes
2 answers
198 views

Proof that every commutative locally compact quantum group arises from a locally compact group

It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof ...
3 votes
1 answer
165 views

quantum invariants, ribbon Tannakian duality and classification of ribbon Hopf algebras

In a nutshell, my question is: Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction? I will now make it more precise. One could define a ...
3 votes
0 answers
107 views

How to get $U(N)_k$ Kac-Moody modules and characters from $N \cdot k$ Dirac Fermions using $U(N \cdot k)_1 / SU(k)_N$?

It is known that the $U(N)_k$ Kac-Moody algebra can be written as the coset $U(N)_k = U(N \cdot k)_1 / SU(k)_N$. (This fact is related to the level-rank duality of $U(N)_k \leftrightarrow U(k)_N$.) A ...
4 votes
0 answers
165 views

Coloured Jones polynomial at 4th root of unity and Arf invariant

Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
5 votes
1 answer
154 views

Explicit correspondence between classical double and quantum double

Proposition 12.3 of Etingof and Schiffmann's "Lectures on Quantum Groups" states the following claim. Proposition 12.3. Let $H$ be a quantized enveloping algebra and let $\mathfrak{g}$ be ...
55 votes
2 answers
11k views

What is quantum algebra?

This might be a very naive question. But what is quantum algebra, really? Wikipedia defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a ...
7 votes
0 answers
331 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 [KSZ06]. We are interesting in an alternative ...
4 votes
0 answers
56 views

When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?

Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...
2 votes
0 answers
120 views

Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?

Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})...
6 votes
1 answer
223 views

Does $U_q (\mathfrak{sl}_2)$ have a universal $R$-matrix?

Consider the standard quantum group $U_q (\mathfrak{sl}_2)$ over the field $\mathbb{C}(q)$ of rational functions (or over $\mathbb{C}$ if $q \in \mathbb{C}$ is not a root of unity), with the usual ...
7 votes
0 answers
151 views

How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?

Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity. Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...
9 votes
0 answers
381 views

Finite-dimensional representations of quantum $SU(2)$

The most famous of all the quantum groups is $SU_q(2)$ - the Quantum special unitary group. The irreducible comodules of this quantum group are very well understood - they are labelled by integers (or ...
3 votes
1 answer
180 views

A filtration on Drinfeld-Jimbo quantum enveloping algebras

For the universal enveloping algebra $U(\frak{g})$ of a Lie algebra $\frak{g}$, one can define in a natural way an increasing $\mathbb{N}_{0}$-filtration. By the Poincaré-Birkhoff–Witt theorem, the ...
2 votes
0 answers
28 views

Examples of (weak-)bialgebras/Hopf algebras with a finite dimensional unitary representation and corepresentation and polynomial growth rate

I need examples (the more the better, even better if there is a systematic way of construction) of (weak-)bialgebras or (weak-)Hopf algebras $H$ with a finite dimensional representation $\rho$ and a ...
0 votes
0 answers
105 views

A variant of quantum harmonic oscillators

We have the following variant of harmonic oscillators. $$ \left\{ \begin{array}{**lr**} T = a + a^\dagger\\ a | n \rangle = \sqrt{[n]} |n-1 \rangle \\ a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
2 votes
0 answers
103 views

Morphism of discrete quantum groups

In the paper Kazhdan's Property T for Discrete Quantum Groups , we read the following fragment: First, note that I think there is a typo and that codomain and domain of the dual maps have to be ...
3 votes
1 answer
190 views

Quantum group associated to a reductive group

In most of the classical references about quantum groups, these objects are defined as a one-parameter deformation of the universal enveloping algebra. However, I have read in several papers that it ...
4 votes
1 answer
175 views

Drinfeld-Jimbo quantum groups for $q=0$

In the Wikipedia page of Drinfeld--Jimbo quantum groups the values of $q=0,1$ are excluded so as to avoid dividing by zero. The $q=1$ case is discussed in this old question. What about the $q=0$ case? ...
11 votes
2 answers
1k views

Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has ...
1 vote
1 answer
80 views

Show that if $V\in M(B_0(H)\otimes A)$, then $V(B(H)\otimes 1)V^*\not\subseteq B(H)\otimes A$ where $A$ is a specific unital $C^*$-algebra

Let $\mathbb{G}$ be a compact (quantum group) with function algebra $(C(\mathbb{G}), \Delta)$ and Haar state $\varphi_{\mathbb{G}}$. Consider the associated GNS-representation $\pi_{\mathbb{G}}: C(\...
8 votes
1 answer
390 views

What is the ring structure on Lusztig's integral form of quantum $\mathfrak{sl}(2)$?

Consider the quantum group $U_q(\mathfrak{sl}_2)$, with generators $E,F,K$ such that $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$. Write $[n]=\frac{q^n-q^{-n}}{q-q^{-1}}$, and $[n]!=[n][n-1]\dotsm[1]$. In ...
8 votes
0 answers
233 views

$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$. Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
2 votes
0 answers
80 views

The dual of elements $E$, $F$, and $H$ of $U_h(\mathfrak{sl}_2)$ corresponds to which element of $F_h(\mathrm{SL}_2)$ by isomorphism?

$\newcommand{\sl}{\mathfrak{sl}}\DeclareMathOperator\SL{SL}$Let $U_h(\sl_2)$ be the quantized universal enveloping algebra of $\sl_2(\mathbb{C})$ and $F_h(\SL_2)$ be the quantized function algebra of $...
6 votes
1 answer
282 views

Quantum exterior algebra

In Generalisation of the quantum exterior algebra the quantum exterior algebra is discussed: $$ K\langle x_1,\dotsc x_n\rangle/(x_i^2,x_i x_j + q_{i,j}x_j x_i), $$ with nonzero field elements $q_{i,j}...
5 votes
0 answers
207 views

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 ...
1 vote
0 answers
122 views

How to make sense of $\mathrm{Mat}_q(n \times n)$? Are there notions of quantum vector space, quantum linear algebra, etc?

Given some algebra $\mathcal{A}$ and $q \in \mathbb{C}$, we say that a matrix $M \in \mathrm{Mat}(n \times n ; \mathcal{A})$ is a quantum matrix in $\mathrm{Mat}_q(n \times n)$ iff the following ...
5 votes
1 answer
239 views

Completely isometric coaction of discrete quantum group is multiplicative?

Let $\mathbb{G}$ be a compact quantum group (in the sense of Woronowicz) with discrete dual $\widehat{\mathbb{G}}$ which we view as a von Neumann algebraic locally compact quantum group (in the sense ...
24 votes
3 answers
3k views

How many definitions are there of the Jones polynomial?

Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
3 votes
1 answer
116 views

Unitary in adjointable operators associated with equivariant Hilbert module

Consider the following fragment from the article "Tannaka–Krein duality for compact quantum homogeneous spaces. I. General theory" by De Commer and Yamashita: How exactly is $\mathcal{E}\...
4 votes
1 answer
167 views

Reference request: decomposability of $\mathbb{G}$-Hilbert modules

Let $\mathbb{G}$ be a compact quantum group, $B$ be a $C^*$-algebra together with a right action $$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $*$-homomorphism satisfying $(\beta \...
7 votes
0 answers
385 views

How to define $U_q \mathfrak{g}$ without generators and relations?

I'm trying to learn something about quantum groups. The related definitions tend to consist of formulas which are not extremely intuitive, on the first glance. So I wonder how the amount of formulas ...
3 votes
0 answers
109 views

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) ...
53 votes
4 answers
5k views

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 ...
7 votes
1 answer
751 views

Is there a good differential calculus for quantum SU(3)?

For quantum $\operatorname{SU}(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $\operatorname{SU}(2)$ by $a$, $b$, $c$, $d$, then the ideal of $\ker(\epsilon)$...
6 votes
1 answer
217 views

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 ...
2 votes
1 answer
216 views

Relation between the modular categories SU(2)_n and Sp(n)_1

The online database [1] provides a list of some modular tensor categories classified by rank. Let us consider the two modular categories denoted kmA1_$\ell$ and kmC$\ell$_1 (i.e. Kac Moody $A_1$ level ...

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