**50**

votes

**4**answers

4k 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 ...

**34**

votes

**5**answers

3k 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 ...

**32**

votes

**4**answers

1k 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 ...

**25**

votes

**6**answers

2k 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 ...

**24**

votes

**2**answers

868 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 ...

**24**

votes

**3**answers

1k views

### What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?

In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of $U_q\...

**24**

votes

**1**answer

1k 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, ...

**23**

votes

**2**answers

2k 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 $...

**23**

votes

**2**answers

854 views

### Why should affine lie algebras and quantum groups have equivalent representation theories?

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra $\mathfrak{...

**21**

votes

**1**answer

2k 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 ...

**21**

votes

**6**answers

1k views

### Is there a q-analog to the braid group?

The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups:
$$ 1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$
where $S_n$ is the symmetric ...

**21**

votes

**1**answer

421 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 ...

**20**

votes

**0**answers

570 views

### Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...

**19**

votes

**4**answers

1k 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 ...

**18**

votes

**11**answers

5k views

### Introduction to deformation theory (of algebras)?

So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...

**18**

votes

**9**answers

1k 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 ...

**17**

votes

**1**answer

2k 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 ...

**17**

votes

**3**answers

6k 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 ...

**17**

votes

**2**answers

2k 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" ...

**17**

votes

**1**answer

727 views

### 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 ...

**17**

votes

**1**answer

2k views

### Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups

Which $6j$-symbols for quantised enveloping algebras are known explicitly?
The quantum $6j$-symbols for $sl(2)$ are well-known. The references are
Masbaum and Vogel and
Frenkel and Khovanov.
What ...

**17**

votes

**1**answer

410 views

### Is the representation category of quantum groups at root of unity visibly unitary?

Let $\mathfrak g$ be a simple Lie algebra.
By taking the specialization at $q^\ell=1$ of a certain integral version¹ of the quantum group $U_q(\mathfrak g)$,
and by considering a certain quotient ...

**17**

votes

**2**answers

997 views

### Does the super Temperley-Lieb algebra have a Z-form?

Background Let V denote the standard (2-dimensional) module for the Lie algebra sl2(C), or equivalently for the universal envelope U = U(sl2(C)). The Temperley-Lieb algebra TLd is the algebra of ...

**16**

votes

**0**answers

269 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 ...

**15**

votes

**3**answers

2k views

### Quantum group as (relative) Drinfeld double?

The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E_i$ and $F_i$ corresponding to the simple ...

**14**

votes

**2**answers

631 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(\...

**14**

votes

**1**answer

459 views

### Categorifying the equality of product and coproduct of symmetric functions

Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...

**13**

votes

**2**answers

758 views

### Where does the canonical basis differ from the KLR basis?

The question implicitly asked in Ben Webster's question is: Does the canonical basis of Uq(n+) agree with the basis coming from categorification via Khovanov-Lauda-Rouqier algebras?
Thanks to ...

**13**

votes

**1**answer

452 views

### $\text{Rep}(D(G))$ as representation category of a vertex operator algebra

The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as ...

**13**

votes

**1**answer

543 views

### A left inverse for the comultiplication on a Hopf von Neumann algebra

Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments.
$\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following ...

**13**

votes

**1**answer

713 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 ...

**12**

votes

**1**answer

488 views

### $(q,x)$-analog of $n!$

While doing some work in geometric representation theory I have come across the following
sequence of polynomials in two variables $(q,x)$ which I would like to denote
by $n!_{q,x}$. For small $n$ ...

**12**

votes

**3**answers

964 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 ...

**12**

votes

**0**answers

315 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 ...

**11**

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 ...

**11**

votes

**3**answers

1k views

### Quantum group Uq(sl(2))

I'm looking at motivating the standard deformation of $U(\mathfrak{sl}(2))$. As an algebra $U(\mathfrak{sl}(2))$ is generated by $X,Y$ and $H$ and subject to the relations $[X,Y] = H$, $[H,X] = 2X$ ...

**11**

votes

**2**answers

531 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(\...

**11**

votes

**2**answers

663 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 ...

**11**

votes

**3**answers

1k 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, ...

**11**

votes

**1**answer

498 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}...

**11**

votes

**1**answer

313 views

### Compact Quantum Groups and the Existence of the Classical Haar Measure

Before I state my question, let me provide the definition of a compact quantum group.
Definition: An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called a compact quantum group if
$...

**11**

votes

**1**answer

507 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 ...

**11**

votes

**1**answer

494 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 ...

**11**

votes

**1**answer

481 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 ...

**11**

votes

**0**answers

409 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 ...

**10**

votes

**2**answers

1k views

### What is the relation between quantum symmetry and quantum groups?

What kind of role do quantum groups play in modern physics ?
Do quantum groups naturally arise in quantum mechanics or quantum field theories?
What should quantum symmetry refer to ?
Can we say that ...

**10**

votes

**2**answers

186 views

### Symmetries of module categories over the category of representations of quantum $sl(2)$

The category $\mathcal{C}_l$ of tilting modules of the quantum group $U_q(sl_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when ...

**10**

votes

**1**answer

582 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 ...

**10**

votes

**0**answers

203 views

### Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...

**10**

votes

**0**answers

433 views

### canonical basis via Gelfand-Tsetlin basis

Do there exist explicit formulas for the action of Lusztig's canonical basis of
$U_q(\mathfrak n_+)$ in the Gelfand-Tsetlin basis of a Verma module for $sl(n)$?