Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory

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### Comodules of the $B,C$ and $D$ series quantum groups

In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...

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**1**answer

258 views

### Does a nonabelian Picard group exist?

Over a noncommutative algebra $A$ we have no problem in defining invertible bimodules (as in the book by Bass on algebraic $K$-theory) - corresponding to line bundles over topological spaces $X$ if $A=...

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115 views

### Are the weight spaces of indecomposable $U_q\mathfrak{sl}(2)$-modules at most 2-dimensional?

This is a follow up of this question.
Let $U_q\mathfrak{sl}(2)$ be Lusztig's integral form of the quantized enveloping algebra of $\mathfrak{sl}_2$, specialised at $q$ a root of unity. This is an ...

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**1**answer

119 views

### Crystal basis for quantum groups and Lie algebras

Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\...

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73 views

### Are the quantum groups $C_q[SU_{1,1}]$ and $C_q[SL_{2}(R)]$ isomorphic?

Classically the group of Moebius transformations of the unit disk and Moebius transformations of the upper half plane are isomorphic, as the unit disk and upper half plane are transformed into each ...

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32 views

### Simple modules of quantum toroidal algebras

Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials.
...

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242 views

### What is the (quasi-) classical limit of categorified quantum groups?

$\newcommand{\g}{\mathfrak g}$
Let $G$ be a reductive group and $U_q(\g)$ the associated quantum group. One can argue that the classical limit of $U_q(\g)$ is $G$ or $\g$, with some Poisson structure, ...

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270 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 ...

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126 views

### Deformation quantization of Poisson bracket without star-product

Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...

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80 views

### How to write a braiding as a matrix?

Let $V$ be the vector representation of $sl_n$. Then $V \otimes V$ is a $U_q(sl_n)$-module. Suppose that a braiding
\begin{align}
\Psi: V \otimes V \to V \otimes V
\end{align}
satisfies the ...

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66 views

### Bialgebraic structure of Sklyanin algebra

Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...

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**1**answer

79 views

### Comodules of Cosemisimple Hopf Algebras

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...

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**1**answer

90 views

### When are Morita classes represented by certain structured algebra objects?

Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ...

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468 views

### Computing in quantum groups

I'd be interested in doing some computations in quantum groups $ U_q(\mathfrak g)$ that are conceptually simple (``is this element 0"?, and $\mathfrak g = sl_5$), but are somewhat lengthy to do by ...

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105 views

### When are the braid relations in a quasitriangular Hopf algebra equivalent?

Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}...

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**1**answer

109 views

### The Quantum Group ${\cal O}_q(SL(n))$, for $q>1$

For the quantum group ${\cal O}_q(SL(n))$, $q\in \mathbb{R}$, I have read, without a proof, that for $p>1$, there exists a $q\in (0,1)$ such that
$$
{\cal O}_p(SL(n)) \simeq {\cal O}_q(SL(n)).
$$
...

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**1**answer

94 views

### Definition of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...

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91 views

### Modular double of elliptic quantum group

By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as $\mathfrak{...

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43 views

### How to value $\Omega$ in T-system for twisted quantum affine algebras?

Let us proceed to the unrestricted T-systems. Choose $h\in {\mathbb{C}\backslash 2\pi \sqrt{-1} \mathbb{Q}}$ arbitrarily.
The unrestricted T-system for $U_{q}(X_{N}^{(\mathfrak{k})})$ is the following ...

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86 views

### Self-dual vertex algebras

Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute
$$
...

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**1**answer

216 views

### Understanding “Decategorified” symplectic Khovanov homology

In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...

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**1**answer

199 views

### Bounding $p$-adic characters and Jacquet-Langlands tranfert

I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_p$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on $...

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**1**answer

127 views

### How to obtain an quantization of the algebra of functions of a given space?

My question is inspired by quantum spheres and quantum projective spaces (studied by Dijkhuizen & Noumi, Vaksman & Soibelman, etc., i.e. "A FAMILY OF QUANTUM PROJECTIVE SPACES AND RELATED $q$-...

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83 views

### Distinguishing the Duflo star product

$\newcommand{\g}{\mathfrak g}\newcommand{\h}{\hbar}$
For a finite dimensional Lie algebra $\g$, he Duflo isomorphism is a complicated algebra isomorphism between the $\g$-invariant part $S(\g)^\g$ of ...

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144 views

### Quantization of $S^2$ as $C^*$-algebra?

The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...

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143 views

### Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...

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**1**answer

106 views

### Classifying Low Dimensional Solutions of the Yang--Baxter Equation

What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions?
To make my question more specific, have all solutions for dimension $2$ and $3$ been ...

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**1**answer

90 views

### Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by $...

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185 views

### Flat connection from gauged WZW model

$\newcommand{\g}{\mathfrak g}$
$\newcommand{\h}{\mathfrak h}$
In short my question is :
Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ?
Some ...

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82 views

### Cosemi-simple FRT Hopf Algebras

This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...

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183 views

### How to make a premodular category a modular tensor category?

A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the ...

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1k views

### Why are quantum groups so called?

I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...

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**1**answer

230 views

### Open questions on (finite) tensor categories

I would like to know about problems on (finite) tensor categories. I have read Etingof´s notes from his course at MIT. I have a question:
There exists any reference where I can find an open problem ...

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**1**answer

125 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 ...

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144 views

### When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?
It seems that one can relatively easy ...

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**1**answer

131 views

### 6j symbols of SU(4) at level 4

Does anybody know of a reference that gives the (quantum) 6j symbols of SU(4) at level 4?
Alternatively, I know the S-matrix and the fusion rules, in the form
$a \times b = \sum_i N^{ab}_{c_i} c_i$
...

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**2**answers

273 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,...

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124 views

### An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see here.
We are interesting in an alternative ...

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**2**answers

227 views

### Appropriate Recursion relations for Wigner 3j Symbols

I am attempting to code the Cosmic Microwave Lensed Temperature and Polarisation power spectra from first principles and have been told to code the relevant Wigner 3j symbols using recursion rather ...

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**1**answer

143 views

### What are the applications of the depth 2 reduction to the subfactors theory?

Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth $2$ (...

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73 views

### The role of the Vandermonde determinant in representations of affine Lie algebras

I am reading a paper 'Yangians and R-matrices' by Chari & Pressley (1990) and to classify representations for particular quantum groups, they define a "quantum Vandermonde determinant". They also ...

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57 views

### Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...

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38 views

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

Since I got a very good answer to my previous question,
217585,
I am asking a sequel by moving up a level.
Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and ...

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**1**answer

110 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 ...

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147 views

### What's the relation between half-twists, star structures and bar involutions on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...

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42 views

### Changing the sign in the definition of the cocommutator of a coboundary Lie bialgebra

A Lie bialgebra is a Lie algebra additionally equipped with a 1-cocycle $\delta: {\mathfrak g}\to \Lambda^2 {\mathfrak g}$ that satisfies the co-Jacobi identity. Non-trivial Lie bialgebras can be ...

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**1**answer

171 views

### Kernel of the natural map between group $C^*$-algebras

Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...

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421 views

### Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?

The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...

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73 views

### Dense Hopf $*$-subalgebra of compact quantum groups and cancellation laws

Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital $*$-...

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**1**answer

61 views

### Radicals of co-quasitriangular map

Let $B=\mathcal{O}_R\left(GL(n)\right)$ be a localization of the algebra $A(R)$ of functions on the quantum formal group corresponding to the matrix $R$ ["Quantization of Lie groups and Lie algebras", ...