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

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### The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...

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148 views

### Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?

This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...

**31**

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**5**answers

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### Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...

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**2**answers

667 views

### quantum groups… not via presentations

Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a_{ij}$, the quantum group $U_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K_i$, $E_i$, $F_i$ (the ...

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**1**answer

212 views

### $q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?

Define n-quantum vector space to be the algebra
$$
{\mathbb C}_q^n := \mathbb{C}< x_i ~ | ~ i =1, \ldots, N>/<x_i x_j = q x_j x_i ~ | ~ i< j>.
$$
For $q=1$, we get the usual polynomial ...

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**4**answers

3k views

### Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...

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**3**answers

372 views

### What is a Homotopy between $L_\infty$-algebra morphisms II

I would like to continue on question I asking, what is a homotopy between
Lie infinity algebras, since I'm not satisfied in two directions:
1.) The naive approach to define a homotopy would be ...

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**2**answers

359 views

### Derangements and q-variants

Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ derangements of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( ...

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**0**answers

73 views

### Asymptotics of the quantum exponential

Let $\epsilon$ be an $N$th root of unity, and $q=\epsilon e^h$ where $h<0$.
I am trying to give a derivation of the lead term of
$$(z;q)_{\infty}=\prod_{n=1}^{\infty}(1-zq^n),$$ as $h\rightarrow ...

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votes

**1**answer

277 views

### Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.
Questions: are there definitions of image and kernel for a ...

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89 views

### Clebsch Gordan coefficients of compact quantum groups

Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...

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**2**answers

460 views

### The category of subfactors extending the category of groups?

This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...

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votes

**1**answer

746 views

### Non-“weakly group theoretical” integral fusion categories?

Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices?
$\small{\begin{smallmatrix}
1 & ...

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votes

**1**answer

223 views

### A linear category with objects of infinite length but which is otherwise finite?

Fix a ground field $k$. By a linear category I will mean an Abelian category which is compatibly enriched over $k$-vector spaces. A linear category is called finite if it satisfies the following four ...

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119 views

### Deformation quantization of a closed Riemann surface with genus >1

Quantization of of an elliptic curve can be done in different ways.
In C^*-algebraic version,
one can start with the C^*-algebra ...

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**3**answers

1k views

### What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...

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**1**answer

225 views

### Does the limit in the Volume conjecture converge?

The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...

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**11**answers

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### What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...

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**0**answers

145 views

### Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...

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**1**answer

944 views

### Spectral theory for self-adjoint field operators on a symmetric Fock space

Background
Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F(H):= \mathbb{C} ...

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**1**answer

162 views

### Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...

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**1**answer

369 views

### When are Jones-Wenzl projectors defined?

(I am hoping that someone well-versed in the literature of Temperley-Lieb algebras or of quantum groups at roots of unity can answer my question. Fingers crossed.)
Consider the Temperley-Lieb algebra ...

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127 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 ...

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144 views

### Understanding a program for computing Khovanov homology

I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp
The ...

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**0**answers

84 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 ...

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**1**answer

302 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 ...

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**0**answers

142 views

### On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...

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276 views

### Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background:
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...

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**1**answer

254 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 ...

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**1**answer

573 views

### Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer ...

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128 views

### Are there workable numerical approaches for the pentagon equation?

Warning: this post is the "numerical" analog of
Are there workable algebraic geometry approaches for the pentagon equation?
I've replaced "algebraic geometry" by "numerical" in its content,
...

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551 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 ...

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### A few questions about Kontsevich formality

[K] refers to Kontsevich's paper "Deformation quantization of Poisson manifolds, I".
Background
Let $X$ be a smooth affine variety (over $\mathbb{C}$ or maybe a field of characteristic zero) or ...

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**2**answers

191 views

### How weird can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?
The reason I am interested in this is that my ...

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**1**answer

172 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.

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**1**answer

133 views

### classification of irreducible finite dimensional representation of affine hecke algebra of type A

Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity.
The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...

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votes

**1**answer

183 views

### Universal ribbon category of ribbon graphs

I'm skimming through Turaev's "Quantum invariants of knots and 3-manifolds". One of the main results is Theorem 2.5. In my opinion, this Theorem is conceptually half-baked: 1) The ribbon structure on ...

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### Deformation quantization and quantum cohomology (or Fukaya category) — are they related?

Good afternoon.
Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of ...

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**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 ...

**11**

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**1**answer

137 views

### Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories

Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all ...

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**2**answers

344 views

### Jones polynomial of the concatenation of two braids

Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings.
Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$?
Here, ...

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328 views

### How unique are extensions of TQFTs to lower dimension?

Say I have an "ordinary" TQFT $F$ of dimension $n$, assigning groups or vector spaces to closed $(n-1)$-manifolds and linear maps to cobordisms. Consider the different ways $F$ can be obtained from a ...

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122 views

### Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra?
For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...

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**6**answers

1k views

### What is an algebraic group over a noncommutative ring?

Let $R$ be a (noncommutative) ring. (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.) Then I think I know what "linear ...

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**1**answer

155 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 ...

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214 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 ...

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184 views

### Where is the Courant operad discussed?

Where is the Courant operad discussed? And hopefully defined precisely.
By the Courant operad or rather a suitable generalization of operad to accommodate the inner product, the operad whose ...

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votes

**1**answer

137 views

### Geometric Intuition of $P^+$ in Modular Tensor Categories

I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...

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**1**answer

86 views

### Equivalence of star products on two differents Poisson algebras?

Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...

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**1**answer

1k views

### The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results:
- Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980).
- A Galois correspondence for depth 2 irreducible subfactors ...