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

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16
votes
0answers
273 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 ...
15
votes
0answers
182 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
0answers
596 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 ...
13
votes
0answers
603 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 ...
12
votes
0answers
265 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, ...
12
votes
0answers
316 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 ...
12
votes
0answers
461 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 $\pi_1(\Sigma)$. There ...
12
votes
0answers
720 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 ...
11
votes
0answers
410 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 why ...
11
votes
0answers
370 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
votes
0answers
537 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 $q\in k$...
9
votes
0answers
230 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. ...
8
votes
0answers
74 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 ...
8
votes
0answers
144 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 ...
8
votes
0answers
272 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 ...
8
votes
0answers
539 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 $0\leq k\leq n/2$. \begin{align} P_{0,0}&=1\\ \text{for $n\geq 1$}...
7
votes
0answers
188 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 ...
7
votes
0answers
150 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
votes
0answers
171 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
0answers
153 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
0answers
634 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
0answers
193 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
0answers
116 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 ...
6
votes
0answers
94 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{...
6
votes
0answers
146 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 ...
6
votes
0answers
125 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 ...
6
votes
0answers
148 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 ...
6
votes
0answers
193 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 Etingof-...
6
votes
0answers
310 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
0answers
205 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
0answers
286 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
votes
0answers
434 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
0answers
376 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
0answers
129 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 ...
5
votes
0answers
84 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 ...
5
votes
0answers
145 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
0answers
253 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
0answers
326 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
0answers
462 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
0answers
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
0answers
232 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 ...
5
votes
0answers
283 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
0answers
330 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
0answers
135 views

Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?

A fusion ring is a finite dimensional complex space $\mathbb{C}\mathcal{B}$ together with a distinguished basis $\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j = \sum_k n_{ij}^kh_k $...
4
votes
0answers
147 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 ...
4
votes
0answers
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 ...
4
votes
0answers
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 $*$-...
4
votes
0answers
99 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
0answers
64 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
0answers
114 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 0^...