**0**

votes

**0**answers

10 views

### Convolution on finite weak Hopf $C^*$-algebra

Let $\mathbb{A}$ be a finite weak Hopf $C^*$-algebra, and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{A}}$ and the convolution $a * b = ...

**7**

votes

**0**answers

117 views

### When is Rep(U_q(g)) invariant under q -> -q and why?

Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...

**2**

votes

**0**answers

65 views

### Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$

I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167:
Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra ...

**18**

votes

**3**answers

499 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 ...

**18**

votes

**1**answer

286 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 ...

**6**

votes

**1**answer

159 views

### Abel's five terms relation from Yang-Baxter equation?

Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations?
If yes, how?
Thanks for any help.

**4**

votes

**1**answer

110 views

### PBW basis and canonical basis

Consider the example of $\mathfrak{g} = sl_3$. Then
$$
\mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-},
$$
where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, ...

**3**

votes

**2**answers

240 views

### Algebraic Groups, Modules, and Comodules

Background:
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H} := \text{Alg}_k\{H; k\},
$$
we recall (see Abe Chapter 4 ...

**0**

votes

**0**answers

41 views

### Does the standard Podlés sphere have a quasitriangular Hopf algebra structure? Do quantum homogeneous spaces have one in general?

Function spaces on (classical) homogeneous spaces can have a bialgebra structure:
Take $S^2$ to be the unital, associative algebra generated by $x, y, z$ with the relation $x^2 + y^2 + z^2 = 1$ and ...

**9**

votes

**1**answer

217 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
...

**2**

votes

**1**answer

149 views

### “Quantum Littlewood-Richardson” Rule?

Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...

**11**

votes

**0**answers

191 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 ...

**2**

votes

**1**answer

96 views

### What is the multiplicative unitary for SU_q(2) (or other quantum groups)?

Consider a (von Neumann algebraic) locally compact quantum group $(M, \Delta, \phi, \psi)$ where the von Neumann algebra $M$ is realized as operators on the Hilbert space $H$. There is a ...

**3**

votes

**0**answers

77 views

### States and extremal states of quantum SU(2) and the PodleÅ› sphere

Is there any description (preferably somehow related to the original generators) for the state space (as in C*-algebras) of quantum SU(2) and the PodleÅ› sphere? If so (this is pushing my luck) are the ...

**10**

votes

**1**answer

274 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 ...

**1**

vote

**0**answers

64 views

### What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?

What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum ...

**3**

votes

**0**answers

66 views

### Is multiplication continuous in quantum-SU(2) with respect to the $L^2$-norm

For the Hopf $\ast$-algebra ${\cal O}_q(SU(2)$, with its unique Haar measure $h$, we have an inner product, and a norm, on ${\cal O}_q(SU(2))$ defined by
$$
<x,y> : = h(xy^*), ...

**3**

votes

**3**answers

319 views

### When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?

Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...

**16**

votes

**1**answer

237 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 versionÂ¹ of the quantum group $U_q(\mathfrak g)$,
and by considering a certain quotient ...

**2**

votes

**0**answers

135 views

### Canonical basis of quantum groups

I am trying to understand the canonical basis of quantum groups and different ways to construct the canonical basis of quantum groups. In the comments of Lusztig's papers, the paper [92], CANONICAL ...

**2**

votes

**0**answers

115 views

### quantum deformations of tensor category

I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...

**4**

votes

**2**answers

176 views

### Reference request: “duality” relations between $U_q(\mathfrak{g})$, $O_q(G)$ and $O_q(G^*)$

Let $\mathfrak{g}$ be a bialgebra, $\mathfrak{g}^*$ its dual, and $G$ and $G^*$ the corresponding connected simply-connected Poisson-Lie groups. I have repeatedly heard claims of the following ...

**1**

vote

**0**answers

64 views

### Reference request: Lusztig's symmetries

Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$,
\begin{align}
...

**3**

votes

**0**answers

112 views

### How a unitary corepresentation of a Hopf C*-algebra, deals with the antipode?

Let $\mathcal{A}$ be a Kac algebra (Hopf C*-algebra), with comultiplication $\delta$, counit $\epsilon$ and antipode $S$.
The following definition comes from this paper (p51-52) of ...

**4**

votes

**1**answer

188 views

### Towards a quantum version of Schur's orthogonality relations

This is taken from Timmermann's Invitation to Quantum Groups and Duality.
Hi folks I am struggling a little with a small calculation in the above text.
I will just get right into it.
Lemma 3.2.5
...

**15**

votes

**0**answers

361 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 ...

**0**

votes

**1**answer

83 views

### dual quantum plane

Let $k\left\{\phi, \gamma\right\}$ be free algebra and $I_{q}$ be the two sided ideal generated by the elements $\phi \gamma + q^{-1}\gamma \phi$, $\gamma^{2}$ and $\phi^{2}$, $k_{q}[\phi,\gamma]= ...

**4**

votes

**0**answers

82 views

### Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras

As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...

**3**

votes

**1**answer

152 views

### example of a compact quantum group at a root of unity?

In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some ...

**1**

vote

**0**answers

83 views

### Quantum Algebras — Crystal Basis/Graph

Suppose I have a finite-dimensional irreducible $U_q(sl_2)$-module say $V$, and (L,B) is its crystal basis.
How do you find the crystal basis of the evaluation $U'$-module $V_{x=1}$? And is there a ...

**5**

votes

**1**answer

226 views

### Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...

**1**

vote

**0**answers

68 views

### Classifying Equivariant Maps Between Fin-Dim Irreducible Modules

Let $G$ be a compact semi-simple Lie group, (or to be even more concrete let $G = SL(N)$), and let $V$ and $W$ be finite dimensional irreducible representations of $G$. Surely it is very well-known ...

**6**

votes

**1**answer

298 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 ...

**4**

votes

**2**answers

284 views

### lie algebras, Kac Moody, and quantum mechanics book

Hi all, I've just finished a graduated course on Kac-Moody algebras, and I'm really looking for some reading in regard to their applications to Quantum Mechanics. Can you help?

**6**

votes

**1**answer

272 views

### Are Turaev--Viro invariants secretly a discretized path integral?

Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with ...

**0**

votes

**0**answers

175 views

### polynomial representation of $sl_{2}(k)$

Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
...

**11**

votes

**2**answers

523 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 ...

**3**

votes

**0**answers

216 views

### Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types

Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible ...

**2**

votes

**1**answer

175 views

### Hopf Duals and Matrix Coefficients

One defines the finite dual of a Hopf algebra $A$ as
$$
H^o := \{f \in H^* ~|~ f(I) = 0, \text{ for some ideal $I$ of $H$ with } \dim_C(H/I) < \infty \}.
$$
As is well-known, $H^o$ has a ...

**1**

vote

**1**answer

109 views

### Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations

The representations of the quantized enveloping algebra $U_q(\frak{sl}_n)$ are labeled by the positive weight lattice $P^+$ of the classical Lie algebra $\frak{sl}_n$. Moreover, the dual Hopf algebra ...

**4**

votes

**2**answers

231 views

### Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation
$$
[E,F] = \frac{K-K^{-1}}{q-q^{-1}}.
$$
To address this problem, one has ...

**3**

votes

**0**answers

108 views

### Hopf Algebra Pairings and Module-Comodule-Equivalences

Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...

**0**

votes

**0**answers

130 views

### $h$-adic Completion of $U_q(\frak{sl}_2)$?

Consider the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as ...

**4**

votes

**4**answers

290 views

### Non-Drinfeld--Jimbo Deformations and Finite Quantum Groups

As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called ...

**7**

votes

**1**answer

166 views

### Real forms of Drinfeld-Jimbo quantum groups

A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and ...

**4**

votes

**2**answers

439 views

### $q$-Deforming Woronowicz's Leibniz Rule

The Woronowicz definition of a differential calculus over an algebra consists of a pair $(\Omega,$d$)$, where $\Omega$ is an $A-A$-bimodule, and
$$
\text{d}:A \to \Omega,
$$
is a bimodule map, ...

**4**

votes

**1**answer

136 views

### Question about unusual highest weight modules for $U_q(sl(2))$

Background
Let $U_q(sl(2))$ be the quantum group associated with $sl(2)$ i.e. the associative algebra with 1 over $Q(q)$ generated by $x^+,x^-,K,K^{-1}$ with relations
$$KK^{-1}=K^{-1}K=1$$
...

**1**

vote

**0**answers

223 views

### Why are there two Hopf algebra structures on a Kac--Moody Algebra.

For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by
$$
\Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ...

**1**

vote

**1**answer

110 views

### Zero Sums in a $q$-Deformation Remain Zero for $q=1$

Looking at previous question, I begun to think and came upon the following more solid question: Let $SU_q(N)$ be the usual quantized coordinated algebra of $SU(N)$. Now consider, for some fixed value ...

**7**

votes

**0**answers

149 views

### Does the braid group act faithfully on the quantized enveloping algebra?

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where ...