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.

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Cohomological interpretation of the equivalence between a quantization of a Lie bialgebra and a QUE algebra

In section 6.2 on quantization Chari and Pressley describe an interesting cohomological interpretation of the similarity of a quantization of a Lie bialgebra and a QUE algebra. For that they started ...
2 votes
1 answer
87 views

Understanding definition of quantization of a Poisson-Hopf algebra

I am going through the chapter Quantization of Lie bialgebras from the book A Guide to Quantum Groups by Chari and Pressley. There I found a notion called Quantization which deals with deformations of ...
6 votes
2 answers
124 views

Where does the univeral $R$-matrix of $U_q(\mathfrak g)$ live?

Let $\mathfrak g$ be a complex simple Lie algebra and let $U_q(\mathfrak g)$ denote the Drinfeld-Jimbo quantum group associated to $\mathfrak g$. I will assume that $U_q(\mathfrak g)$ is a $\mathbb C(...
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3 votes
0 answers
116 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 ...
6 votes
0 answers
231 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 ...
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2 votes
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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)^{\...
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2 votes
0 answers
97 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 ...
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4 votes
1 answer
239 views

Relation between TQFT representations and factorizable sheaves

I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks. More ...
3 votes
0 answers
76 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) ...
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5 votes
1 answer
163 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 ...
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1 vote
1 answer
126 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 ...
0 votes
0 answers
60 views

Associativity of Quantum Double

Here is the statement about the associativity of the quantum double of bialgebras in Klimyk-Schmudgen "Quantum Groups ..." (Sec 8.2.1) Can anyone help me derive the formula of on bottom of ...
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66 views

Quantum double construction of a quantum group

Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$ with a Cartan subalgebra $\mathfrak t$ and Borel subalgebras $\mathfrak b_\pm \subset \mathfrak g$. (You can assume $\mathfrak g=sl(2)$ ...
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1 vote
0 answers
429 views

New (?) math object. Looking for (if existing) literature [closed]

I am interested in any literature about the following mathematical property. Let $V$ a vector space and $G$ a group acting on $V$. What is the name for the property of a set of operators $H=\{h:V\to V\...
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3 votes
1 answer
179 views

Relating different definitions of dual of a compact quantum group

Let $\mathbb{G}$ be a compact quantum group in the sense of Woronowicz. We can look at its associated dense Hopf$^*$-subalgebra $\mathbb{C}[\mathbb{G}]$. Hence, in the framework of multiplier Hopf $*$-...
  • 405
7 votes
1 answer
235 views

Does the category of commutative and cocommutative Hopf algebras have enough injectives?

It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see https://arxiv.org/abs/1502.04001v2 and its references). But does it have enough injectives? What about ...
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3 votes
1 answer
210 views

Motivating quantum groups from knot invariants

Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...
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4 votes
0 answers
120 views

Lusztig's root datum

In his "Introduction to quantum groups", Lusztig first defines a Cartan datum $(I,\cdot)$ and then a root datum of $(I,\cdot)$-type, which suggests (to me) that a root datum is not ...
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5 votes
1 answer
120 views

Exactness of functors in a $C^*$-tensor category

I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. In section 2.3 "Fiber functors and reconstruction theorems*, the following ...
  • 405
5 votes
0 answers
106 views

Product of $U^+_q(\mathfrak{sl}_2)_i$ in $U_q(\mathfrak{g})$ according to some reduced expression

Let $\mathfrak{g}$ be some simple Lie algebra, $\alpha_1,\alpha_2,\cdots,\alpha_n$ be its simple roots. Let $U^+_q(\mathfrak{sl}_2)_i$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $E_i,K_i$. ...
2 votes
1 answer
218 views

Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]

Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as $\operatorname{Rep}(G)$ ...
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6 votes
1 answer
130 views

Norm of contragredient of unitary representations of compact quantum groups

Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms. Let $G = (A, \Delta)$ be a ...
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1 vote
0 answers
79 views

How to understand a definition in KLR algebra in the setting of quantum affine algebras?

I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra: $$ X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1) $$ This ...
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4 votes
1 answer
210 views

$\text{Mod}(A)$ is an $E_n$ category $\Leftrightarrow$ $A$ is an ??? algebra

Say we're working in a symmetric monoidal $\infty$-category $\mathcal{S}$, and $A$ is an associative algebra in it. For instance, $$\mathcal{S}\ =\ \text{dg vector spaces},\ \ \ A\ =\ \text{a dg ...
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1 vote
0 answers
113 views

Representation of quantum groups

Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if ...
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7 votes
1 answer
244 views

Is the comultiplication of a compact quantum group always injective?

Let $(A, \Delta)$ be a compact quantum group in the sense of Woronowicz. Is it true that the comultiplication $\Delta : A \to A \otimes A$ always injective? This is true for both the universal (...
4 votes
1 answer
103 views

Exponential map and Lie correspondence within a Hopf algebra setting

The Cartier-Konstant-Milnor-Moore (et al.) theorem for Hopf algebras states that a cocommutative Hopf algebra over $\mathbb{C}$ is isomorphic to a smash product of a universal enveloping algebra of a ...
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2 votes
1 answer
116 views

Action of a group $G$ induces a coaction on $C_0(G)$

In this question, I follow the book "An invitation to quantum groups and duality" by Timmerman, p259. Let $G$ be a locally compact group and $C$ be a $C^*$-algebra. Assume an action $$\alpha:...
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3 votes
1 answer
96 views

Norm antipode on a Kac-type compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group. Consider the associated dense Hopf$^*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ ...
  • 405
3 votes
1 answer
171 views

Woronowicz characters are multiplicative

I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. Let $G$ be a $C^*$-algebraic compact quantum group with function algebra $(C(G), \Delta)...
  • 405
10 votes
0 answers
193 views

Quantum groups at small roots of 1

I wonder if there is any literature about representations of quantum groups at a root of 1 of small order. For example, I would like to understand the case of $\mathrm{SL}(2)$ and $q=-1$ (in the ...
4 votes
0 answers
139 views

Tensor product of representations on a compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$. Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
  • 405
5 votes
1 answer
164 views

Subrepresentations of C*-algebraic compact quantum groups

Let $\mathbb{G}$ be a compact quantum group with function algebra $(C(\mathbb{G}), \Delta)$ (in the sense of Woronowicz). Let $X \in M(B_0(H) \otimes C(\mathbb{G}))$ be a (possibly infinite-...
  • 405
2 votes
1 answer
131 views

Timmerman's "An invitation to quantum groups and duality" corollary 3.2.8

Consider the following propositions of Timmerman's book "An invitation to quantum groups and duality": I do not understand the equivalence $$h(\mathfrak{C}(\delta_V)S(\mathfrak{C}(\delta_W))...
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3 votes
1 answer
192 views

Characterising algebraic compact quantum groups among Hopf $^*$-algebras

Let $(A, \Delta)$ be a Hopf $^*$-algebra. Assume that $\{u^\alpha\}_{\alpha \in I}$ is a maximal collection of pairwise inequivalent irreducible unitary corepresentation matrices. I want to show that $...
  • 405
4 votes
2 answers
125 views

If $\operatorname{Hom}(\delta_V, \delta_W) = 0$, then $\mathfrak{C}(\delta_V) \cap \mathfrak{C}(\delta_W) = 0.$

Let $(A, \Delta)$ be a Hopf $^*$-algebra and $\delta_V: V \to V \otimes A$ and $\delta_W: W \to W \otimes A$ be two corepresentations of $(A, \Delta).$ Assume that the space of intertwiners $\...
  • 405
1 vote
1 answer
80 views

Orthogonality relations for Haar state and antipode (Timmerman)

Consider the following proposition from Timmerman's "An invitation to quantum groups and duality": I am having trouble seeing why the boxed equations are true (Note that on the left the ...
  • 405
2 votes
0 answers
105 views

Anticommutation of convolution products on trace class operators of quantum groups

This question was originally posted to MathStackExchange. Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
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5 votes
2 answers
304 views

Classifying Hopf algebras that admit a single irreducible comodule

Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule $$ k \to k \otimes H, ~~ k \mapsto k ...
4 votes
1 answer
462 views

Name for a Hopf algebra whose only grouplike element is the identity?

For a $k$-Hopf algebra $H$ and element $h \in H$ is called grouplike is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1_k$ ($\epsilon$ is the counit). The identity $1_H$ is clearly grouplike, but in ...
4 votes
1 answer
113 views

Hopf algebra that is unimodular and counimodular but not involutory

I'm looking for a finite-dimensional Hopf algebra (over any field) that is unimodular, has unimodular dual, but is not involutory. Is there such a thing? Here's what I know: By Radford's formula, the ...
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5 votes
0 answers
108 views

Classification of connected finite affine type A crystals

In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...
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3 votes
1 answer
95 views

Nonstandard Podles spheres as $U_c(\frak{h})$ invariants

In this paper Podles introduced a $2$-parameter family of $q$-deformed spheres $S_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "...
3 votes
1 answer
219 views

The adjoint representation of $U_q({\frak sl}_2)$ on itself

Let $U_q(\frak{sl}_2)$ denote the quantum universal enveloping algebra of $\frak{sl}_2$, and consider the adjoint action $$ \mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(...
10 votes
2 answers
653 views

What is the discrete quantum group associated to a compact group?

Let $G$ be a compact topological group. Then $G$ is a CQG with function algebra $C(G)$ and the usual comultiplication on $C(G)$. Is there an easy description of the dual discrete quantum group $\...
  • 405
12 votes
1 answer
323 views

Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators

$\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E_\pm, H$ with commutation relations $$ [H,E_{\pm}]=\pm 2 E_\pm,\quad [E_+,E_-]=H $$ admits the well-known representation on $\mathbb{C}[x]$ with $$ ...
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5 votes
0 answers
208 views

Induction for quantum group

I am confused about a claim in the article Representation of quantum algebras by Henning Haahr Andersen, Patrick Polo and Wen Kexin. I probably misunderstood a definition, but I found two claims about ...
9 votes
2 answers
405 views

Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?

In a Lie algebra $\mathfrak{g}$ the Jacobi identity $\newcommand{\bracket}[2]{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$ holds. ...
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6 votes
1 answer
126 views

Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity

I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$ works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
3 votes
0 answers
90 views

Fiber product of group rings

Let $K$ be a field. For a group $G$ we write $K[G]$ for the group ring of $G$. Given group homomorphisms $F \to G, H \to G$ is the canonical map $$ K[F \times_G H] \to K[F] \times_{K[G]} K[H] $$ an ...

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