**22**

votes

**6**answers

1k 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

1k 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

711 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 ...

**3**

votes

**3**answers

752 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

516 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 ...

**16**

votes

**2**answers

949 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

160 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

316 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

753 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

362 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

433 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

434 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

423 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

298 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 ...