**4**

votes

**0**answers

278 views

### Working with quadratic Lie algebras

A quadratic Lie algebra is a Lie algebra with an invariant inner product and the main examples are semisimple Lie algebras. This definition then makes sense in any linear symmetric monoidal category. ...

**5**

votes

**1**answer

535 views

### Weyl Character Formula for Quantum Groups

How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of ...

**6**

votes

**0**answers

217 views

### Explicit Braid Group Reps from quantum SO(N) at roots of unity

This question is related to this one (and indeed the goals are similar).
Let $N$ be odd and consider the braided fusion category $\mathcal{C}$ (actually modular) obtained from $U_q\mathfrak{so}_N$ ...

**2**

votes

**1**answer

143 views

### Formula for the Matrix Elements of the Inverse of special linear Universal R-Matrix of Uq(sln)

Motivated by this question, I'm going to ask a much more direct one: Let $R$ be a universal R-matrix for the quasitriangular Hopf algebra ${\cal U}_q ({\mathfrak sl}_N)$, $~$ $R ^{-1}$ its inverse, ...

**2**

votes

**1**answer

220 views

### Reference for the Hecke relation for the universal R-matrix

I've come across a reference in a paper to the
Hecke relation for the universal R-matrix of a quasi-triangular Hopf algebra.
I've looked around, standard references, online etc, but can't seem ...

**10**

votes

**1**answer

445 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 ...

**1**

vote

**2**answers

225 views

### Group and Hopf Algebra Structures for Projective Varieties

Let $V$ be a projective (or affine) variety. Does there exist a bijective correspondence between group structures on $V$ and Hopf algebra structures on the coordinate ring of $V$?

**13**

votes

**3**answers

1k 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 ...

**3**

votes

**0**answers

128 views

### The Tangent Spaces of the Bicovariant Calculi over Quantum SU(3)

In Woronowicz's approach to differential calculi on Hopf algebras, a calculus over an algebra $A$ can be specified by a finite dimensional subspace of the dual of $A$. The best known example is his ...

**3**

votes

**2**answers

457 views

### Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules

Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 ...

**6**

votes

**1**answer

453 views

### Is there a good reference for the relationship between the Yangian and formal based loop group?

For every finite dimensional semi-simple Lie group $\mathfrak{g}$, we have a loop algebra $\mathfrak{g}[t,t^{-1}]$. This loop algebra has a natural invariant inner product by taking the residue at ...

**5**

votes

**2**answers

192 views

### What is the explanation for the special form of representations of three string braid group constructed using quantum groups information supplied

It is well-known that representations of quantised enveloping algebras give representations of braid groups. For the examples that I know explicitly the representations of the three string braid group ...

**3**

votes

**0**answers

122 views

### Restricting Differential Calculi to Quotients

Let $A$ be an algebra, $B \subset A$ a subalgebra, and $(\Omega^1(A),d)$ a first order differential calculus over $A$. Now, as is well known, the subalgebra of $\Omega^1(A)$ consisting of elements of ...

**1**

vote

**0**answers

184 views

### Bicovariant Calculi on the Quantum Unitary Groups

The bicovariant differential calculi on quantum-$SU(n)$ have been classified (by Schmudgen I think) and have been shown to have non-classical dimension. My question is whether or not the bicovariant ...

**3**

votes

**3**answers

337 views

### Group-Adjoint and Hopf-Algebra-Adjoint Maps

I've reading some introductory quantum group material and am trying to understand the algebra-space correspondence in the classical case. One object I'm stuck on is the adjoint coaction
$$
Ad_R: a ...

**6**

votes

**2**answers

836 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 ...

**15**

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 ...

**7**

votes

**2**answers

839 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

319 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

670 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 ...

**3**

votes

**1**answer

223 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 ...

**13**

votes

**11**answers

4k 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 ...

**5**

votes

**1**answer

248 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
...

**16**

votes

**1**answer

1k 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

863 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

572 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

448 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

510 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

544 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 ...

**2**

votes

**1**answer

122 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

357 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

261 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

304 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 ...

**21**

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

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

678 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

727 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

486 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 ...

**15**

votes

**2**answers

900 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

156 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

303 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

727 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

350 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

421 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

404 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 ...

**2**

votes

**2**answers

374 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 ...

**3**

votes

**1**answer

287 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 ...