All Questions
Tagged with qa.quantum-algebra quantum-groups
323 questions
67
votes
4
answers
9k
views
Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
- 18.7k
55
votes
2
answers
11k
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 ...
- 5,040
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
44
votes
5
answers
5k
views
Groups, quantum groups and (fill in the blank)
In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
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38
votes
6
answers
4k
views
Why Drinfel'd-Jimbo-type quantum groups?
Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...
- 13k
32
votes
1
answer
5k
views
Which is the correct version of a quantum group at a root of unity?
By this I mean the specialisation of the quantum group Uq(g) with q a root of unity, and the 'correct' meaning of 'correct' (enclosed in quotations since there isn't necessarily a correct answer) is ...
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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
30
votes
2
answers
2k
views
quantum groups... not via presentations
Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a_{ij}$, the quantum group $U_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K_i$, $E_i$, $F_i$ (the ...
- 43.2k
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
26
votes
1
answer
1k
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The Major Families of Quantum Groups
If we define a quantum group to be a quasi-triangular or coquasi-triangular Hopf algebra, then what are the major families of quantum groups?
Of couse, to start with we have the h-adic completions ...
- 1,479
26
votes
1
answer
2k
views
Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?
The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
- 28.1k
24
votes
9
answers
3k
views
expository papers related to quantum groups
Hello all,
I know basic representation theory(finite groups, lie groups&lie algebras) and I want to get a flavor of quantum groups (why they are useful, important results etc) and other related ...
- 1,719
24
votes
3
answers
3k
views
How many definitions are there of the Jones polynomial?
Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
- 18.7k
23
votes
1
answer
3k
views
Grothendieck and Non-commutative Geometry?
When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the ...
- 1,776
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
21
votes
4
answers
2k
views
Why are quantum groups so called?
I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
- 827
21
votes
3
answers
12k
views
Quantum mathematics?
"Quantum" as a term/prefix used to be genuinely physical: what was supposed to be physically continuous turned out to be physically quantized.
What sense does this distinction make inside ...
21
votes
1
answer
863
views
Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?
The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...
- 1,474
18
votes
3
answers
3k
views
How to calculate the Witten-Reshetikhin-Turaev invariants from a triangulation?
I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, ...
- 18.7k
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
18
votes
2
answers
1k
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 $(A, \Delta, \varepsilon, \Phi)$ ...
- 183
18
votes
0
answers
577
views
Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?
Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...
- 118k
17
votes
2
answers
830
views
Relationship between "different" quantum deformations
This is a generic question, a good answer to it may be a reference to a corresponding paper\textbook, but any useful comments would be okay too.
Let $\mathfrak{g}$ be a (simple) Lie algebra and $U_q(\...
- 765
17
votes
3
answers
1k
views
Isomorphisms of quantum planes
Let $k$ be a field and $q\in k^{*}$. The quantum plane $k_{q}[x,y]$ is the algebra $k\langle x,y\rangle/\langle xy=qyx \rangle$ (i.e. the quotient of the free non-commutative $k$-algebra on two ...
- 2,218
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
15
votes
2
answers
1k
views
Quantum-Jimbo Algebras: Why Such Fuss About Roots of Unity?
Coming from a Lie algebraic background, I'm trying to branch onto quantum group theory. The divide I see all the time, is $q$ a root of unity or $q$ not a root of unity. I am wondering why is this? If ...
- 151
15
votes
1
answer
657
views
Is every finite quantum group a quantum symmetry group?
This post is basically a quantum extension of Is every finite group a group of “symmetries”?
Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra.
Frucht's theorem ...
15
votes
0
answers
1k
views
Representations of quantum groups at roots of unity
I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
- 18.7k
14
votes
2
answers
751
views
Projective modules over quantum groups
My question is short:
How can one calculate $\operatorname{Tor}_{U_q(\mathfrak g)}(k,k)$?
($k$ is the ground field of characteristic zero).
If we had a regular universal enveloping algebra $U(\...
- 18.7k
14
votes
0
answers
580
views
Is the quantum dilogarithm related in any way to cohomology of quantum groups?
Is the quantum dilogarithm related in any way to cohomology of quantum groups?
This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain why ...
- 18.7k
13
votes
7
answers
2k
views
Open problems in the theory of compact quantum groups
What are the important open problems in the theory of compact quantum groups? Or conjectures?
Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules ...
13
votes
6
answers
2k
views
Hopf algebras arising as Group Algebras
Every commutative $C^*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is ...
- 1,462
13
votes
2
answers
997
views
Can one define quantized universal enveloping algebras in a basis-free way?
(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&...
- 32.5k
13
votes
1
answer
736
views
The Inverse of a Universal R-Matrix for Quantized Universal Enveloping Algebra of sl2 and the Dual Pairing with SUq(2)
I have recently begun to study quasi-triangular structures and have come across a problem I can't resolve. Let ${\cal U}_q({\mathfrak sl}_N)$ denote the quantised enveloping algebra of ${\mathfrak sl}...
- 1,776
13
votes
1
answer
848
views
Are Turaev--Viro invariants secretly a discretized path integral?
Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with $\left|WRT(M)\...
- 18.7k
13
votes
1
answer
281
views
Finiteness of the number of Hopf subalgebras
Let $H$ be a finite-dimensional Hopf algebra over the complex field.
Question: Does $ H $ have a finite number of Hopf subalgebras?
In the case where $ H $ is semisimple, the answer is yes. According ...
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
$$
...
- 133
12
votes
2
answers
2k
views
Is there any published physics article where $q$-mathematics is applied?
Excuse me for the concern, but I want to ask you a question.
In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science ...
- 551
12
votes
3
answers
832
views
Axiomatic definition of quantum groups
This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group?
There are ...
- 3,318
12
votes
1
answer
723
views
Unitary representations of Quantum Groups
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...
- 8,559
12
votes
1
answer
834
views
R-matrices, crystal bases, and the limit as q -> 1
I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of ...
- 8,559
12
votes
0
answers
285
views
Is there a non-Kac complex finite dimensional semisimple Hopf algebra?
A complex (finite-dimensional) Hopf algebra is said to be a
Kac algebra if it is a ${\rm C^{\star}}$-algebra in such a way that the comultiplication $\Delta$ is a $\star$-homomorphism. Obviously, a (...
11
votes
3
answers
663
views
Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules
In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra:
The space of ...
- 219
11
votes
2
answers
1k
views
Quantized Enveloping Algebras at $q=1$
As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation
$$
[E,F] = \frac{K-K^{-1}}{q-q^{-1}}.
$$
To address this problem, one has ...
- 263
11
votes
2
answers
1k
views
Yang–Baxter explanation
What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? ...
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11
votes
3
answers
1k
views
Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)?
We have known that any finite dim Lie algebra can be embeded into it's enveloping algebra $U(\mathfrak{g})$, my question is: is there any "quantum Lie algebra" embeded into the quantum enveloping ...
- 131
11
votes
2
answers
804
views
q-difference equations and quantum mechanics
I have been trying to understand why the term quantum is so easily accepted for calculus based on q-numbers $[n]_q=\frac{q^n-1}{q-1}$ and q-analogs of classical operators (derivatives, integrals,...).
...
- 982
11
votes
1
answer
556
views
Generators of the Odd Dimensional Quantum Spheres
As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the ...
- 1,453
11
votes
1
answer
629
views
$q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?
Define n-quantum vector space to be the algebra
$$
{\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 ...
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