Skip to main content

All Questions

Filter by
Sorted by
Tagged with
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&...
fusheng's user avatar
  • 137
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 ...
szantag's user avatar
  • 143
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 ...
jeykey's user avatar
  • 31
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 ...
Alvaro Martinez's user avatar
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 ...
Ji Woong Park's user avatar
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 \...
J. De Ro's user avatar
  • 525
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 ...
Jake Wetlock's user avatar
  • 1,144
3 votes
1 answer
492 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 $\operatorname{Rep}(G)$ ...
Meths's user avatar
  • 309
1 vote
0 answers
166 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 ...
Jianrong Li's user avatar
  • 6,201
11 votes
0 answers
252 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 ...
Alexander Braverman's user avatar
5 votes
1 answer
209 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-...
Andromeda's user avatar
  • 175
3 votes
1 answer
367 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_{(...
Jake Wetlock's user avatar
  • 1,144
13 votes
1 answer
411 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 $$ ...
Yamero's user avatar
  • 133
6 votes
1 answer
226 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 ...
pomello gaudente's user avatar
4 votes
3 answers
540 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 ...
Jake Wetlock's user avatar
  • 1,144
5 votes
1 answer
181 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 $$\...
user avatar
4 votes
1 answer
101 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 ...
Piet Bongers's user avatar
0 votes
0 answers
106 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 ...
Jake Wetlock's user avatar
  • 1,144
6 votes
0 answers
442 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 ...
Cubic Bear's user avatar
4 votes
3 answers
344 views

Coinvariants of tensor products of Hopf algebras

Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way. The axioms of Hopf algebras imply that $$ G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
Todd Claymore's user avatar
3 votes
1 answer
104 views

Irreducibility of product bicomodules

Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right, $H$-comodule respectively. The tensor product $$ V \otimes W $$ has an obvious $H$-$H$-bicomodule structure. If $V$ and $W$ are ...
Jake Wetlock's user avatar
  • 1,144
5 votes
2 answers
680 views

Characters on Hopf algebras

For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
Fofi Konstantopoulou's user avatar
8 votes
3 answers
528 views

Classification of $\operatorname{Rep} D(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
Student's user avatar
  • 5,230
4 votes
1 answer
445 views

A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules

Does anyone have a proof for the following Lemma? Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...
cl4y70n____'s user avatar
5 votes
0 answers
287 views

Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$

I am looking for a direct proof that a highest weight representation of $Y(\mathfrak{sl}_2)$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial. The results was ...
Zhihua Chang's user avatar
2 votes
0 answers
108 views

Why are the quantum Fock spaces in FLOTW the same as Uglov's?

Theorem 2.5 in the well-known FLOTW paper [1] and Theorem 2.1 in Uglov's paper [2] both refer to the original JMMO paper [3] to define quantum Fock spaces, i.e. Fock spaces for $U_q(\widehat{\...
Chris Schoennenbeck's user avatar
5 votes
2 answers
403 views

Indecomposable, non-simple, modules of quantum groups at roots of unity

Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...
Konstantinos Kanakoglou's user avatar
11 votes
3 answers
663 views

Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra: The space of ...
Bas Winkelman's user avatar
5 votes
1 answer
497 views

Quantum groups at $q=-1$

For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...
Bas Winkelman's user avatar
32 votes
1 answer
2k views

Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$

Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
John Pardon's user avatar
  • 18.7k
9 votes
2 answers
362 views

Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups

For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
Bas Winkelman's user avatar
18 votes
3 answers
2k views

Hopf dual of the Hopf dual

Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some ...
Nadia SUSY's user avatar
4 votes
0 answers
145 views

Indecomposable modules for the big quantum group

I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
JC Arias's user avatar
5 votes
0 answers
144 views

Are the integral forms of quantized coordinate algebras always Noetherian?

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected ...
Nicolas Dupré's user avatar
7 votes
2 answers
378 views

Hopf Subalgebras of Quantized Algebras

As is well known, quantized enveloping algebras $U_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets ...
Bas Winkelman's user avatar
5 votes
1 answer
183 views

Schur's Lemma for Quantized Universal Enveloping Algebra

Let $U_q(\mathfrak{g})$ (defined over $\mathbb{C}(q)$) be the quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$. Let $M$ a finite-dimensional simple left $U_q(\mathfrak{g})$...
Zhihua Chang's user avatar
9 votes
1 answer
766 views

The difference between $q$-deformations and $h$-deformations

What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras? In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping ...
Mathematician 42's user avatar
7 votes
0 answers
183 views

Relationship between R-matrix and Casimir element?

Given a simple Lie algebra $\mathfrak{g}$, is there any relation between its Casimir element and the $R$-matrix of the related Yangian $Y(\mathfrak{g})$?
Zhihua Chang's user avatar
17 votes
2 answers
2k views

Examples of representations of quantum groups

I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
asv's user avatar
  • 21.8k
1 vote
0 answers
88 views

Reference request: Nichols algebras of a braided vector space with a diagonal braiding

Are there some references of the proof of the following result? Let $(V, c)$ be a braided vector space over a field $k$ with a basis $x_1, \ldots, x_n$, where $c$ is a diagonal braiding given by \...
Jianrong Li's user avatar
  • 6,201
6 votes
1 answer
392 views

Corepresentations of Tensor Products of Hopf Algebras

Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure ...
Abo Kutis-Felan's user avatar
2 votes
0 answers
71 views

Comodules of the $B,C$ and $D$ series quantum groups

In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...
Malcom Stuart's user avatar
7 votes
0 answers
161 views

Are the weight spaces of indecomposable $U_q\mathfrak{sl}(2)$-modules at most 2-dimensional?

This is a follow up of this question. Let $U_q\mathfrak{sl}(2)$ be Lusztig's integral form of the quantized enveloping algebra of $\mathfrak{sl}_2$, specialised at $q$ a root of unity. This is an ...
André Henriques's user avatar
5 votes
1 answer
429 views

Crystal basis for quantum groups and Lie algebras

Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\...
Jianrong Li's user avatar
  • 6,201
3 votes
0 answers
97 views

Simple modules of quantum toroidal algebras

Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials. ...
Jianrong Li's user avatar
  • 6,201
22 votes
0 answers
481 views

What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?

Let $\mathfrak g=\mathfrak{sl}(2)$. Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness. Let $U_q\mathfrak g$ be Lusztig's integral form of the ...
André Henriques's user avatar
6 votes
1 answer
272 views

Bialgebraic structure of Sklyanin algebra

Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...
Kevin Ye's user avatar
  • 367
5 votes
0 answers
191 views

Modular double of elliptic quantum group

By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as $\mathfrak{...
Kevin Ye's user avatar
  • 367
7 votes
2 answers
405 views

The Irreducible Representations of the Sekine Quantum Groups

Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set $$\mathcal{B}_n=\mathbb{Z}_n\times\mathbb{Z}_n=\{(i,...
JP McCarthy's user avatar
  • 1,027
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 ...
asv's user avatar
  • 21.8k