# Questions tagged [quantum-groups]

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 called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.

552 questions
Filter by
Sorted by
Tagged with
67 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 ...
206 views

126 views

148 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 ...
135 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? ...
1 vote
65 views

### Product of matrix entry and representation

Consider a semisimple lie algebra $\mathfrak{g}$ and the corresponding quantum group $\mathcal U_q(\mathfrak{g})$ over $\mathbf{Q}(q)$. Consider two dominant weights $\lambda,\mu\in P^+$, a matrix ...
81 views

### Invariant weights associated to algebraic quantum groups

Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$. ...
93 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 ...
1 vote
75 views

297 views

### Every locally compact group gives rise to a locally compact quantum group

A locally compact quantum group (in the sense of Vaes-Kustermans) consists of the data $(M, \Delta, \varphi, \psi)$ with $M$ a von Neumann algebra, $\Delta: M \to M \overline{\otimes} M$ a normal ...
1 vote
50 views

### Looking for paper ‘Lyndon bases and the multiplicative formula for R-matrices’ by M.Rosso

Does anyone have a link to the paper ‘Lyndon bases and the multiplicative formula for R-matrices’ by M. Rosso? I cannot find it in Google Scholar or Z-Lib. Thank you!
228 views

101 views

164 views

### 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 ...
150 views

314 views
+250

### 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 ...
320 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 ...
80 views

### Canonical basis of the invariant part of $O_q(\mathfrak g)^{\otimes N}$

Let $\mathfrak g$ be a semi-simple Lie algebra (We can assume $\mathfrak g=sl(n)$ for simplicity) and let $O_q(\mathfrak g)$ be the corresponding quantum algebra of functions. Then \$O_q(\mathfrak g)^{\...
182 views

### Drinfeld's "almost cocommutative Hopf algebras"

I'm looking for the following paper (in English): Drinfelʹd, V. G. Almost cocommutative Hopf algebras. Leningrad Math. J. 1 (1990), no. 2, 321–342 I can only find the russian version on the internet ...