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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
93 views

A question about the existence of rational functions

I am reading a paper Representations of shifted quantum affine algebras. I have a question about the existence of a rational function about the remark $4.4$ I'll briefly describe the problem. We let $...
14 votes
1 answer
654 views

Quantum braid group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...
11 votes
0 answers
472 views

Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry

In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry. For graphs this had been an open ...
4 votes
0 answers
136 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 ...
8 votes
1 answer
262 views

Some fusion rings/categories I don't recognize

Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...
4 votes
1 answer
126 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 ...
8 votes
1 answer
744 views

A "concrete" example of a one-sided Hopf algebra

I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition. To be more ...
2 votes
0 answers
68 views

Quantum Schubert cell algebra and quantum odd-dimensional euclidean space

De Concini, Kac, Procesi introduced quantum Schubert cell algebra associated to a complex Lie algebra $\mathfrak{g}$ which is denoted by $\mathcal{U}^{w}_{\epsilon}$ where $w$ is an element of Weyl ...
4 votes
0 answers
159 views

Representations of $C\left(SO_q(n)\right)$

A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
7 votes
0 answers
154 views

Nontrivial examples of locally compact quantum groups

What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
53 votes
2 answers
10k 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 ...
6 votes
1 answer
198 views

Nuclearity of $C(\partial_F \mathbb{G})$

Let $\mathbb{G}$ be a discrete quantum group, and consider the non-commutative Furstenberg boundary $\partial_F\mathbb{G}$ with function algebra $C(\partial_F \mathbb{G}) = I_{\mathbb{G}}(\mathbb{C})$....
7 votes
0 answers
329 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 ...
3 votes
0 answers
44 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
106 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})...
2 votes
0 answers
87 views

Category O for (Yangian) toroidal Lie algebras?

Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote: $$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$ $$g_{[2]}^+ := g \...
6 votes
1 answer
163 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 ...
2 votes
1 answer
220 views

Completely contractive Banach algebra structure on the dual of a Hopf $C^*$-algebra

Let $(A, \Delta)$ be a Hopf $C^*$-algebra, i.e. $A$ is a $C^*$-algebra, and $\Delta: A \to M(A\otimes A)$ is an injective non-degenerate $*$-homomorphism that is coassociative: $$(\iota \otimes \Delta)...
6 votes
0 answers
115 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 ...
8 votes
0 answers
342 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
165 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 ...
8 votes
1 answer
212 views

How does the Tannaka duality work for weak Hopf algebras and fusion categories?

I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...
2 votes
0 answers
23 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 ...
5 votes
1 answer
177 views

In the rep theory of Quantum Double, why does the fusion of 2 "pure fluxes" yield a "pure charge"?

Physicist here, so my notation may be different from standard math notation. For the quantum double $D(G)$ of a group $G$, we may write representations of $D(G)$ in the following way: Consider a ...
3 votes
2 answers
103 views

Does unitarity and modularity constrain fusion multiplicities to be 0,1?

If I have a braided tensor category that's unitary and modular, then how does the unitarity and modularity constrain the fusion multiplicities? I know that if $a,b,c \in ob({C})$ satisfy the fusion ...
0 votes
0 answers
100 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\...
4 votes
1 answer
138 views

Examples of discrete quantum group actions on commutative von Neumann algebras

Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ ...
0 votes
0 answers
88 views

Concrete examples of quantum duality principle

Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
0 votes
0 answers
133 views

Is $[n]_q!$ invertible in $\mathbb C [[h]]\ $?

Consider the Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F, H$ and relations $:$ $$[H, E] = 2 E,\ \ [H, F] = - 2 F,\ \ [E, F] = \frac {q^H - q^{-H}} {q - q^{-...
2 votes
0 answers
96 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
164 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 ...
1 vote
0 answers
101 views

How does $R \equiv 1\ (\text {mod}\ h)\ $?

Definition $:$ Let $H$ be a Hopf algebra. An invertible element $R \in H \otimes H$ is called a coboundary structure on $H$ if $(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$ $(2)$ $R_{21} = R^{-1},$ $(...
1 vote
0 answers
71 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 ...
4 votes
1 answer
156 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? ...
0 votes
0 answers
89 views

How to show that quantum $sl_2 (\mathbb C)$ is a Hopf algebra deformation of $U (sl_2 (\mathbb C))\ $?

The quantum $sl_2 (\mathbb C)$ is the non-commutative, non-cocommutative Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F$ and $H$ with the relations $:$ $$[H, ...
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 ...
2 votes
0 answers
85 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 $\psi$. ...
1 vote
1 answer
78 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(\...
2 votes
0 answers
85 views

Quantum groups as bialgebra cohomology classes

My question below is about how to view the quantum group $U_q(\mathfrak{g})$ as a bialgebra cohomology class. Background: If $A$ is a bialgebra, Gerstenhaber and Schack in Bialgebra cohomology, ...
0 votes
2 answers
123 views

Tensor product of operator values weights (in the theory of locally compact quantum groups)

Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object $$\iota \otimes \psi: (M\overline{\otimes} M)_+ \to \widehat{M_+}.$$ ...
7 votes
1 answer
338 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
216 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
73 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 $...
4 votes
1 answer
313 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
0 answers
52 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!
6 votes
1 answer
243 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}...
1 vote
1 answer
97 views

Problem in understanding a fact about Belavin-Drinfeld triple

A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental ...
5 votes
0 answers
190 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
116 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
220 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 ...

1
2 3 4 5
12