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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0answers
87 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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2answers
282 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 ...
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1answer
437 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 ...
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1answer
76 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 ...
5
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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 ...
3
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1answer
74 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
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1answer
183 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_{(...
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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 $\...
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1answer
280 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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178 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 ...
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2answers
342 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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1answer
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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 ...
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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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Quantum orbit method at roots of unity

Chari and Pressley’s A guide to quantum groups, or the original work by Vaksman and Soibelman in 1989, explains that, similarly to the orbit method which relates quantised coadjoint orbits to unitary ...
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3answers
478 views

Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule

A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion. Now every Hopf algebra $H$ admits a one-dimensional ...
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1answer
174 views

Quantum double vs Quantum group

I am trying to understand what is a difference between a quantum double and a quantum group? I thought these two were the same and now I have trouble figuring out how are they related?
3
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1answer
97 views

quantum affine $gl_2$

There are many sources of the relations and Hopf algebra structure of quantum affine $sl_2$ as a deformed enveloping algebra. However, for an application to integrable systems I need to look at ...
5
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2answers
257 views

Low dimensional noncommutative non-cocommutative Hopf algebras

Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
9
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1answer
273 views

The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$

I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
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Why does Kashiwara define $U_q(\mathfrak{g})$ over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$?

When defining crystal bases, why do we typically view $U_q(\mathfrak{g})$ as an algebra over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$? In Kashiwara's original paper introducing crystal bases, he ...
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Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)

I'm interested in solutions to the Yang-Baxter equation $$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$ that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
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47 views

Superfluous axioms for ribbon Hopf algebra

In his book Foundations of quantum group theory, Majid defines (2.1.10) a ribbon Hopf algebra as a quasi-triangular Hopf algebra $(H, R)$ with a special central element $v \in H$ satisfying (1) $v^2 ...
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1answer
280 views

Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig

"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
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2answers
998 views

Why does Drinfeld Unitarization work?

In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
3
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1answer
107 views

What is the determinant of the R-matrix defining the colored Jones polynomial?

Let $V_n$ be the $(n+1)$-dimensional irreducible representation of $\mathcal U = \mathcal{U}_q(\mathfrak{sl}_2)$, and let $\mathbf R \in \mathcal{U} \widehat \otimes \mathcal{U}$ be the universal $R$-...
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1answer
120 views

A comodule algebra map from a Hopf algebra to itself

Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$-comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (non-identity) algebra map $\sigma:H \...
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283 views

Relation Hopf categories and categorified quantum groups

In the paper Hopf Categories Crane and Frenkel gave a definition of a Hopf category, which they considered as a categorification of a quantum group. Categorifications of quantum groups have later been ...
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1answer
109 views

Morphisms between compact quantum groups

Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
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1answer
105 views

Matrix coefficients of a compact quantum group

Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). Definition: A corepresentation matrix of $(A, \Delta)$ is a matrix $a=(a_{i,j}) \in M_n(A)$ such that $$\...
6
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1answer
306 views

Finite compact quantum groups

Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
5
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1answer
146 views

Relating different constructions of the universal compact quantum group

Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections. ...
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0answers
83 views

Galois descent of a Hopf algebra

In a question here on why the classification of commutative Hopf algebras requires the field to be algebraically closed, a brilliant answer is given discussing the notion of Galois descent. As I ...
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1answer
97 views

Non-cosemisimple duals of pointed Hopf algebras

I take the following quote from an answer to this question A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. The quantized enveloping algebras and ...
5
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1answer
259 views

Invertible elements of the Hopf algebra quantum $SU(2)$

Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see https://en.wikipedia.org/wiki/Compact_quantum_group (Note that on the ...
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0answers
62 views

Hopf algebra antipodes and right left comodule equivalences

Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
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1answer
99 views

Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra

It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra?
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0answers
105 views

Is the category of Yetter-Drinfeld modules abelian?

Is $YD(H)$ the category of Yetter--Drinfeld modules over a Hopf algebra (defined over a field $k$) necessarily abelian? If not then what is the simplest example of a Hopf algebra $H$ for which $YD(H)$ ...
7
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212 views

Conceptual proof of braid group actions on quantum groups

Roughly 1990, Lusztig wrote a series of papers on quantum groups. Perhaps the result that the braid groups acts on $U_q(\mathfrak{g})$ is the proof which is least conceptual. The original paper ...
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102 views

Are there attempts to numerically finding algebraic structures over finite-dimensional vector spaces?

By "algebraic structure" I mean a finite set of linear operators between tensor products of copies of one (or more) finite-dimensional (complex or real) vector spaces, fulfilling a set of ...
4
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1answer
253 views

What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?

For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if $$(R\otimes id)(id\otimes R)(R\otimes id) = (id\otimes R)(R\otimes ...
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1answer
300 views

Peter-Weyl theorem (compact quantum groups)

I'm reading the paper Notes on compact quantum groups. In this paper, the following theorem is proven: Question: Why is the marked equality true?
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1answer
106 views

Direct sum of representations of a compact quantum group

Let $(A, \Delta)$ be a compact quantum group and $\{(H_\alpha, v_\alpha)\}$ be a collection of representations of $A$. That is, $$v_\alpha \in M(B_0(H_\alpha) \otimes A); \quad \quad(\text{id}\otimes \...
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1answer
125 views

Definition intertwiner of representations of compact quantum groups

Before asking my question, let me introduce the relevant terminology. Throughout, let $(A, \Delta)$ be a compact quantum group. Definition: A representation $v$ on the Hilbert space $H$ is an element $...
2
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2answers
161 views

Kernel of intertwiner is invariant (compact quantum groups)

Before asking my question, let me introduce the relevant terminology. Throughout, let $(A, \Delta)$ be a compact quantum group. Definition: A representation $v$ on the Hilbert space $H$ is an element $...
4
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1answer
149 views

Constructing intertwiners between representations of compact quantum groups

Consider the following paper by Van Daele en Maes Notes on compact quantum groups. For convenience of the reader, here is a picture of the relevant section: (1) How is compact operator defined in ...
2
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1answer
80 views

Extending $*$-morphisms to the multiplier algebras

I'm reading the following fragment in the paper "Notes on compact quantum groups": While I'm familiar with the multiplier algebra (constructed via double centralizers) and its universal ...
3
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2answers
211 views

Convolution of functionals on compact quantum group

Let $\mathbb{G}= (A, \Delta)$ be a ($C^*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^*= B(A, \mathbb{C})$ obtains a product $$\omega_1*\omega_2:= (\omega_1\otimes \omega_2) ...
4
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0answers
164 views

Are the finite quantum permutation groups, weakly group-theoretical?

Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...
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168 views

Where can I find Drinfeld's original papers on quantum groups?

Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $U_h(\mathfrak{g})$ be the Drinfeld-Jimbo quantum group, i.e. the $\mathbb{C}[[h]]$-algebra topologically generated by $X_i,Y_i,H_i$ where $1\leq i\...
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0answers
123 views

Frobenius reciprocities

An adjunction of the form $\mathrm{Hom}(A \otimes X, Y) \cong \mathrm{Hom}(X, A^* \otimes Y)$ in a rigid monoidal category is sometimes called Frobenius reciprocity. Is there a result that unifies ...

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