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

**17**

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155 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 ...

**14**

votes

**0**answers

158 views

### Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with.
Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...

**14**

votes

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550 views

### Splitting of homomorphism from cactus group to permutation group

We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...

**11**

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253 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 ...

**11**

votes

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574 views

### Capelli determinant = Duflo ( determinant) - was it known ?

Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$).
I googled a lot ...

**11**

votes

**0**answers

377 views

### Is the quantum dilogarithm related in any way to cohomology of quantum groups?

Is the quantum dilogarithm related in any way to cohomology of quantum groups?
This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain ...

**11**

votes

**0**answers

361 views

### When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...

**11**

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518 views

### Given an algebra, can it be realized as a block of a Hopf algebra?

During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in ...

**10**

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361 views

### Generalization of Witten's computation of the volume of moduli space

Let $\Sigma$ be a Riemann surface, and let $X=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\\!/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$. ...

**10**

votes

**0**answers

669 views

### Is there an “arithmetic cobordism category”?

This question is a clumsy attempt to apply a certain analogy. I hope that if the answer is negative it comes with a clarification of the scope and limitations of the analogy.
Arithmetic topology is ...

**9**

votes

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208 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. ...

**9**

votes

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480 views

### Representations of quantum groups at roots of unity

I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...

**8**

votes

**0**answers

131 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} ...

**8**

votes

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265 views

### Are there only finitely many maximal irreducible amenable subfactors at fixed finite index?

A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
Question: Are there only finitely many maximal irreducible amenable subfactors at ...

**7**

votes

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138 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 ...

**7**

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164 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 ...

**7**

votes

**0**answers

145 views

### Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...

**7**

votes

**0**answers

546 views

### duality between universal enveloping and function algebra for GL(n)

Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular ...

**7**

votes

**0**answers

188 views

### Decomposition of certain projectives for cyclotomic q-Schur algebras

In representation theory, a very popular set of finite dimensional algebras are the $q$-Schur algebras, which are given by looking at the endomorphisms of $V^{\otimes d}$ where $V$ is the standard ...

**6**

votes

**0**answers

169 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 ...

**6**

votes

**0**answers

276 views

### Paving conjecture for Toeplitz matrices

Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...

**6**

votes

**0**answers

202 views

### Category of modules over a coPoisson-bialgebra

Fix a ground commutative ring $k$. A coPoisson-bialgebra is a bialgebra $H$ equipped with a linear mapping $\pi:H \rightarrow H \otimes_k H$ s.t.
$\pi$ is a coLie bracket
$\pi$ is a coderivation
...

**6**

votes

**0**answers

465 views

### Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$.
...

**6**

votes

**0**answers

261 views

### Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?

Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, ...

**6**

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428 views

### Where can I find tables of dual canonical basis vectors?

Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra.
Now presumably this algorithm has been implemented ...

**6**

votes

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345 views

### Quantum polynomial rings and singularities

Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...

**5**

votes

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136 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 ...

**5**

votes

**0**answers

224 views

### How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...

**5**

votes

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305 views

### Is there a version of the 2d cobordism hypothesis for surfaces with non-empty incoming and outgoing boundary?

Question: Is there a condition on an object $x$ of an $(\infty,2)$-category $\mathcal C$ which is equivalent to $x = Z(pt_+)$ for a unique TFT $Z$ from the $(\infty,2)$-category of framed bordisms ...

**5**

votes

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416 views

### Jones Polynomial and Quantum Field Theory

I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...

**5**

votes

**0**answers

131 views

### Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$ ?

Let $n\in\mathbb N$. Let $k$ be a commutative ring. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $x_n$ over $k$. (The multiplication in ...

**5**

votes

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188 views

### deformed Gauss Bonnet formula?

I was reading about Gauss-Bonnet for circles, that integrating the curvature over the circle $\gamma(t)$ leads to the number of times $\gamma'(t)$ rotates around the origin. (The degree of the Gauss ...

**5**

votes

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268 views

### Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...

**5**

votes

**0**answers

325 views

### Quantum Drinfeld-Sokolov reduction of a Whittaker module

Take a Whittaker module $Wh$ of a (finite or affine) semi-simple Lie algebra $\mathfrak{g}$ , and apply the quantum Drinfeld-Sokolov reduction $qDS$ with respect to an $sl(2)$ embedding $\rho:sl(2) ...

**4**

votes

**0**answers

56 views

### Is there a perfect integral fusion category with PFdim = 2 mod 4?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff any $1$-dimensional complex representation of $G$ is trivial.
proof: First if ...

**4**

votes

**0**answers

146 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 ...

**4**

votes

**0**answers

99 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 ...

**4**

votes

**0**answers

191 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 ...

**4**

votes

**0**answers

187 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 ...

**4**

votes

**0**answers

95 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 ...

**4**

votes

**0**answers

180 views

### An embedding theorem for a fusion ring planar algebra?

We first recall the embedding theorem for finite depth subfactor planar algebras:
The planar algebra generated by a (finite depth) subfactor, is embeddable into the planar algebra generated by its ...

**4**

votes

**0**answers

178 views

### Existence of a Kac algebra for a given fusion ring in a particular class

A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., ...

**4**

votes

**0**answers

195 views

### Quantum Drinfeld-Sokolov reduction for a module

There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...

**4**

votes

**0**answers

109 views

### Is there a maximal finite depth infinite index irreducible subfactor?

A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...

**4**

votes

**0**answers

124 views

### FRT construction in the case of super algebras

I'm looking on papers which are talking about the super quantum algebra osp(2|1). I want to understand how one applies the FRT construction in the case of osp(2|1).
Of course there is a super ...

**4**

votes

**0**answers

145 views

### The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...

**4**

votes

**0**answers

292 views

### On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials.
Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...

**4**

votes

**0**answers

302 views

### TQFT and Mapping Class Groups

It is well known how could we get a representation of the mapping class group of a surface S (which I assume compact, connected and orientable), given a TQFT. My question is: is there any reference ...

**4**

votes

**0**answers

221 views

### Braid group action on canonical basis

This is a question about Lusztig's theory of based modules. This theory is elementary but far from easy and is developed in Chapter 27 of "Introduction to quantum groups".
Let $V$ be a highest weight ...

**4**

votes

**0**answers

256 views

### Reshetikhin-Turaev and links with a distinguished component

Hi,
This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question.
Let $T$ be the category whose objects are ...