**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 ...

**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 ...

**8**

votes

**2**answers

998 views

### What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?

In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...

**6**

votes

**1**answer

376 views

### Transmutation versus operads

A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...

**5**

votes

**1**answer

723 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 ...

**4**

votes

**1**answer

233 views

### FTR Quantization for any Subalgebra of $GL(n)$?

As is well known, the quantum groups $SU_q(n)$, amongst others, arise from $R$-matrix solutions of the Yang-Baxter Equation. My question is: For any subalgebra of $GL(n)$, does there exist an $R$-...

**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 ...

**6**

votes

**1**answer

265 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
...

**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 ...

**8**

votes

**6**answers

1k 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 ...

**7**

votes

**1**answer

587 views

### Is there a good differential calculus for quantum SU(3)?

For quantum $SU(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $SU(2)$ by $a,b,c,d$, then the ideal of ker($\epsilon)$ corresponding to this calculus is
$$
&...

**4**

votes

**1**answer

497 views

### Canonical basis for the extended quantum enveloping algebras

I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U_q(\mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody ...

**3**

votes

**3**answers

534 views

### Basis of quantum SU(n)

As is well known, the set
$\{a^ib^jc^k | i,j,k \in \mathbb{Z}\_{\geq 0},k>0\} \cup \{b^lc^md^n | l,m,n \in \mathbb{Z}\_{\geq 0}\}$
forms a basis for quantum $SU(2)$. Does anyone know of a basis ...

**2**

votes

**2**answers

567 views

### Connes v Woronowicz - Cyclic Cohomology v Diff Calculi

Following on from my last two questions link text and link text: Is it correct (and useful) to say that the relationship between Connes' cyclic cohomology approach to de Rham cohomology and Woronowicz'...

**2**

votes

**1**answer

125 views

### Classical Calculi as Universal Quotients

As is well known, every differential calculus $(\Omega,d)$ over an algebra $A$ is a quotient of the universal calculus $(\Omega_A,d)$, by some ideal $I$. In the classical case, when $A$ is the ...

**5**

votes

**2**answers

392 views

### Quantum Frobenius II

In a previous question, I asked how Lusztig's quantum Frobenius generalizes the classical Frobenius map on a variety over a finite field. I was directed to a very interesting paper by Kumar and ...

**2**

votes

**1**answer

273 views

### Basis for Universal Calculus

Can anyone give an explicit basis of the universal (noncommutative) differential calculus over an algebra $A$ with basis ${e_i}$. (The universal calculus over $A$ is the kernel of the multiplication ...

**3**

votes

**1**answer

319 views

### Categorifying the group representations

I've heard about this construction on the lecture about higher representation theory:
Given a Lie algebra $g$, one constructs $\mathcal A$, a category whose $K_0$ is the universal enveloping ...

**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 ...

**9**

votes

**5**answers

2k views

### Solutions of the Quantum Yang-Baxter Equation

I am interested in finding non-constant solutions to the following Yang Baxter equation
$$R_{12}(x/y) R_{13}(x/z) R_{23}(y/z) = R_{23}(y/z)
R_{13}(x/z) R_{12}(x/y)$$
where $R(x)$ is an endomorphism ...

**6**

votes

**5**answers

781 views

### Braided Monoidal 2-categories with duals

Which categorifications give explicit braided monoidal 2-categories with duals?
This question is in response to Ben Webster's questions in recent history. The point is that given a braided monoidal 2-...

**4**

votes

**3**answers

812 views

### Quantum Frobenius

In what sense does Lusztig's quantum Frobenius, defined on a quantum enveloping algebra, generalise the classical Frobenius mapping on a variety over a finite field?

**6**

votes

**5**answers

583 views

### Is there a version of Temperley-Lieb using sl(3) rather than sl(2)?

This question is a spin-off from Sammy Black's question on super Temperley-Lieb. Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the ...

**17**

votes

**2**answers

1k 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 ...

**0**

votes

**2**answers

182 views

### Antipode for quantum matrices.

Am I right in assuming that one cannot define an antipode for $M_q(n)$ the bi-algebra of $nXn$ quantum matrices? If so, does anyone know a proof?

**6**

votes

**2**answers

344 views

### How do quantum knot invariants change when I pick a funny ribbon element?

So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ...

**3**

votes

**4**answers

789 views

### An inner product that makes the R-matrix unitary

So, if you talk to the right people, they will tell you that the braiding of the category of representations of a quantum group are not unitary and that one can fix this by taking a different commutor ...

**2**

votes

**3**answers

378 views

### What's the best reference for actual formulas for RT invariants?

If one really wants to understand the formulas for how to construct the Reshetikhin-Turaev 3-manifold invariants coming from quantum groups in terms of R-matrices and such, what's the best reference ...

**3**

votes

**3**answers

461 views

### What is a formula for the “group-like Drinfeld element”?

Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...

**4**

votes

**2**answers

490 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 ...

**4**

votes

**2**answers

492 views

### How does one think about the “off-diagonal” part of the $R$-matrix?

The universal $R$-matrix of a quantized universal enveloping algebra is typically written as the product of two terms, one only involving elements of the Cartan, and one only involving elements of the ...

**3**

votes

**1**answer

321 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 ...