**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\...

**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 ...

**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, ...

**4**

votes

**2**answers

386 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? ...

**4**

votes

**1**answer

319 views

### A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $

Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...

**5**

votes

**1**answer

132 views

### Quantum group representations from (convolution) matrix units?

Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...

**51**

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 ...

**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 ...

**24**

votes

**2**answers

876 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 ...

**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

**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" ...

**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, ...

**16**

votes

**0**answers

274 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 ...

**7**

votes

**2**answers

934 views

### Is there a quantum Hermite reciprocity?

It is well known that there is an isomorphism of $SL_2=SL(V)$ representations
$$
Sym^n(Sym^m(V))\simeq Sym^m(Sym^n(V))
$$
called Hermite reciprocity (discovered in 1854).
My question is: Is there ...

**5**

votes

**1**answer

127 views

### Compact Quantum Groups and FRT-Algebras

As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which ...

**2**

votes

**0**answers

183 views

### The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{...

**7**

votes

**2**answers

276 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,...

**5**

votes

**1**answer

264 views

### Convex PBW bases

Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, $\...

**1**

vote

**1**answer

45 views

### Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules

There are four versions of compatibility conditions of Yetter-Drinfeld modules (left-left, left-right, right-left, right-right) in the article. Are there some references which derive these ...

**8**

votes

**1**answer

195 views

### Brauer-Picard for a fusion category coming from a quantum group

In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A \...

**6**

votes

**1**answer

378 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 ...

**2**

votes

**0**answers

33 views

### Partially permutative matrices

Let $V$ be a finite dimensional vector space over a field $K$. Then a map
$L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\...

**2**

votes

**1**answer

115 views

### Deforming the category of representations of a simple Lie algebra?

This is a follow up question to the insightful answer by Theo Johnson-Freyd to the question 105221.
This answer explained that the quantised enveloping algebra, $U_q(\mathfrak{g})$, (defined by a ...