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

**4**

votes

**2**answers

413 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 ...

**5**

votes

**1**answer

103 views

### What homomorphisms $G \to BrPic(\mathcal{C})$ correspond to group-theoretical $G$-extensions of $\mathcal{C}$?

For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$.
...

**0**

votes

**0**answers

64 views

### How to generalize multiplication and addition to cyclic subfactors?

Let $(N \subset M)$ be a finite index irreducible subfactor and $P=P(N \subset M)$ its subfactor planar algebra.
Definition: $(N \subset M)$ is cyclic if its lattice of intermediate subfactors ...

**2**

votes

**0**answers

199 views

### Is there a non-trivial maximal Hopf algebra?

Let $H$ be a Hopf algebra over an algebraically
closed field $\mathbb{K}$ of characteristic $0$.
Maximal means without left coideal subalgebra $I$ (i.e. $\Delta(I) \subset H \otimes I$) other than ...

**10**

votes

**1**answer

363 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 ...

**9**

votes

**2**answers

346 views

### $6j$-symbols for $U_q({\mathfrak{sl}}_n)$ and colored HOMFLY polynomials

Explicit expression of quantum $6j$-symbolos for $U_q({\mathfrak{sl}_2})$ have been known due to the work of Kirillov and Reshitikhin.
My Question:
How much are known about quantum $6j$-symbolos ...

**1**

vote

**0**answers

62 views

### Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$.
Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...

**2**

votes

**0**answers

129 views

### How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?

**2**

votes

**0**answers

115 views

### Generalising the $L^2$-Norm to Compact Quantum Groups

For a compact quantum group (in the sense) of Woronowicz, we have a noncommutative $C^*$-algebra replacing ${\mathbb C}(G)$. This $C^*$-algebra is endowed with a linear functional generalizing the ...

**1**

vote

**0**answers

127 views

### The planar algebra generated by the biprojections

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two irreducible finite index subfactors.
Let $\mathcal{B}_i$ be the set of all the biprojections of $\mathcal{P}_{2+}(N_i \subset M_i)$.
Let ...

**0**

votes

**0**answers

66 views

### Are the Standard Quantum Groups Coordinate Rings Noetherian?

Are the standard quantum groups $C_q[G]$ Noetherian and if so what is a standard reference?

**8**

votes

**0**answers

108 views

### Comparing a Chevalley basis with the canonical basis of the adjoint module?

First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} ...

**3**

votes

**1**answer

261 views

### Jones polynomial of tangles using Temperley-Lieb algbra?

The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...

**3**

votes

**0**answers

50 views

### Signs associated to self-dual simple objects in a fusion category

Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element:
The square of $a$ equals Muger's "squared dimension" of $X$, an ...

**4**

votes

**1**answer

183 views

### Verlinde Formula and Theta Function Identities

The paper Fusion rules and modular transformations in 2D conformal field theory by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just ...

**5**

votes

**1**answer

452 views

### exceptional cases in Kazhdan-Lusztig

The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine).
What's special about those cases?

**4**

votes

**0**answers

135 views

### Number of Irreducible Representations of $U_q(n)$ of Dimension $n$?

For quantum group $U_q(n)$, is it true that it has exactly two non-isomorphic irreducible corepresentations with dimension $n$, and that one is the dual of the other? I know result is in the Chapter ...

**0**

votes

**1**answer

129 views

### The coproducts $\mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ and $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$

A coproduct $\varphi: \mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ is given by: $x \mapsto 1 \times x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}_q[U]$. There is a ...

**0**

votes

**0**answers

38 views

### References about reality of minimal affinizations of quantum affine algebras

Let $U_q(\widehat{\mathfrak{g}})$ be the quantum affine algebra associated to a complex simple Lie algebra $\mathfrak{g}$. A simple module $M$ of $U_q(\widehat{\mathfrak{g}})$ is called real if $M ...

**2**

votes

**1**answer

103 views

### Is there a tangle encoding the fusion rules?

Let $(N \subset M)$ be an irreducible finite index depth $n$ subfactor. Let $P = P(N \subset M)$ its planar algebra.
Let $(B_i)$ be the finite sequence of $N$-$N$-bimodules appearing in the principal ...

**8**

votes

**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 ...

**1**

vote

**0**answers

124 views

### 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 ...

**9**

votes

**0**answers

196 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**

votes

**5**answers

3k views

### 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 ...

**22**

votes

**2**answers

704 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 ...

**45**

votes

**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 ...

**5**

votes

**3**answers

389 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 ...

**14**

votes

**2**answers

364 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( ...

**4**

votes

**0**answers

84 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 ...

**2**

votes

**1**answer

374 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 ...

**1**

vote

**0**answers

111 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 ...

**4**

votes

**2**answers

469 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 ...

**12**

votes

**1**answer

789 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 & ...

**10**

votes

**1**answer

246 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 ...

**1**

vote

**2**answers

140 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 ...

**17**

votes

**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 ...

**8**

votes

**1**answer

247 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
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...

**47**

votes

**11**answers

5k views

### 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 ...

**5**

votes

**0**answers

159 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 ...

**11**

votes

**1**answer

956 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} ...

**2**

votes

**1**answer

177 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 ...

**12**

votes

**1**answer

407 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 ...

**7**

votes

**0**answers

133 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 ...

**0**

votes

**0**answers

162 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 ...

**2**

votes

**0**answers

95 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 ...

**6**

votes

**1**answer

312 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

**0**answers

165 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 ...

**6**

votes

**2**answers

314 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 & ...

**5**

votes

**1**answer

273 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 ...

**6**

votes

**1**answer

607 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 ...