All Questions
69 questions
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 ...
- 21.8k
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 $N/n\to\alpha\in(0,1)$, ...
- 18.7k
26
votes
2
answers
3k
views
When does Lusztig's canonical basis have non-positive structure coefficients?
I've heard asserted in talks quite a few times that Lusztig's canonical basis for irreducible representations is known to not always have positive structure coefficents for the action of $E_i$ and $...
- 44.7k
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 ...
- 43.2k
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 ...
- 835
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 ...
- 21.8k
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
$$
...
- 133
12
votes
1
answer
723
views
Unitary representations of Quantum Groups
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...
- 8,559
11
votes
3
answers
663
views
Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules
In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra:
The space of ...
- 219
11
votes
1
answer
629
views
$q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?
Define n-quantum vector space to be the algebra
$$
{\mathbb C}_q^n := \mathbb{C}\left< x_i \mid i =1, \ldots, N\right>/\left<x_i x_j = q x_j x_i \mid i<j\right>.
$$
For $q=1$, we get ...
- 465
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 ...
- 7,037
10
votes
1
answer
871
views
Is there a good reference for the relationship between the Yangian and formal based loop group?
For every finite dimensional semi-simple Lie group $\mathfrak{g}$, we have a loop algebra $\mathfrak{g}[t,t^{-1}]$. This loop algebra has a natural invariant inner product by taking the residue at ...
- 44.7k
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 ...
- 219
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 ...
- 395
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 ...
- 1,144
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,...
- 5,230
8
votes
2
answers
819
views
Are there interesting monoidal structures on representations of quantum affine algebras?
Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
- 45.7k
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 ...
- 492
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 ...
- 219
7
votes
1
answer
672
views
The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
I asked this question on Math.Stack but have not had any answers.
Question
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
The trivial ...
- 1,037
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,...
- 1,037
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 $Y(\mathfrak{g})$?
- 671
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 ...
- 43.2k
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 ...
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 ...
- 159
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?...
- 367
6
votes
1
answer
298
views
Are there elements of fixed weight in a crystal not killed by too many Kashiwara operators?
I've come across an annoying lemma trying to finish up an argument, and I was hoping one of you guys knew about it.
Question: Given
a weight $\lambda$ of a simple Lie algebra $\mathfrak g$, and
...
- 44.7k
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 ...
- 719
5
votes
4
answers
532
views
Non-Drinfeld–Jimbo deformations and finite quantum groups
As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called Drinfeld--...
- 263
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 ...
- 7,746
5
votes
1
answer
498
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 ...
- 219
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(\...
- 6,201
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 ...
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-...
- 175
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
$$\...
user167952
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})$...
- 671
5
votes
1
answer
404
views
What is the "right" hermitian structure on tensor products of quantum group representations?
This is pretty specific, but there are some experts around.
So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a ...
- 44.7k
5
votes
1
answer
600
views
exceptional cases in Kazhdan-Lusztig
The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine).
What's special about those cases?
- 43.2k
5
votes
1
answer
951
views
Does the canonical basis of a tensor product of quantum group representations span the isotypic components of tilting modules?
It's a theorem of Lusztig that if one takes two (or more) representations of a quantum group at a non-root-of-unity choice of $q$, and look at the canonical basis of their tensor product, then each ...
- 44.7k
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 ...
- 323
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 ...
- 671
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 ...
- 126
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{...
- 367
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\} = \...
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 ...
- 1,144
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
...
- 145
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 \...
- 525
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 $\...
- 353
4
votes
1
answer
280
views
Vertex embeddings of quantum groups via quivers
Let Uq be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of Uq, we see that for each vertex of the finite Dynkin diagram, there is an inclusion Uq($\hat{sl_2}$)...
- 8,866
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 ...
- 143