Examples when quantum $q$ equals to arithmetic $q$
First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.In the world of quantum mathematics, the letter $q$ is a standard...
View ArticleIs there a non-pointed simple integral modular fusion category?
The weakly group-theoretical conjecture, which suggests a negative response to [ENO11, Question 2], is formulated as follows:Statement 1 (open): Every integral fusion category is weakly...
View ArticleIs there a non-split super-modular positive integral fusion category?
We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let...
View ArticleAlternative to Kontsevich formality
Has anyone considered an alternative approach to Kontsevich formality in which the DGLA of poly-vector fields is deformed to an $L_\infty$-algebra?Some vocabulary:DGLA = Differential Graded Lie...
View ArticleMonoidal class vs gauge class vs Grothendieck class
In the comments under this post, three notions of equivalence classes of unitary modular tensor categories are brought up. They are monoidal classes, gauge classes, and Grothendieck class. Could...
View ArticleIs the Drinfeld element of a semisimple quasitriangular Hopf algebra...
Let $A$ be a finite dimensional semisimple quasitriangular Hopf algebra over $\mathbb{C}$ with universal $R$-matrix denoted by $\mathcal{R}\in A\otimes A$. The Drinfeld element of $A$ is defined...
View Articlequantum invariants, ribbon Tannakian duality and classification of ribbon...
In a nutshell, my question is:Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction?I will now make it more precise. One could define a...
View ArticleWhat algebras generate polynomial count varieties as their representations...
Consider for example commutative polynomial algebra $C[x,y]: xy=yx$, look on $F_p$-matrices satisfying that relation - the number over $F_p$ will be given by polynomial in $p$ (classical result due to...
View ArticleIs there an integral fusion category of the Ising type?
In [EGNO, Section 8.27.3], we read:Any braided fusion category ${\mathcal C}$ is obtained from a weakly anisotropiccategory (namely, the core of ${\mathcal C}$) using finite groups (via...
View ArticleProof that every commutative locally compact quantum group arises from a...
It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof...
View ArticleVertex operator algebras and modular fusion categories
Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C} = \text{Rep}(\mathcal{V})$ be the tensor category of $\mathcal{V}$-modules. It is a conjecture by Vaughan Jones whether every...
View ArticleAffiliating the whole algebra of 'coordinates' with a locally compact quantum...
When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$)...
View ArticleRepresentations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where...
View ArticleNon weakly-group-theoretical integral fusion category
Is there an integral fusion category of rank $7$, FPdim $210$ and type $(1,5,5,5,6,7,7)$ with the following fusion rules (or the little $\color{purple}{\text{variation}}$...
View ArticleProof of redundancy for defining relation in current algebra $J$ presentation
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The $\mathfrak{g}$-current (Lie) algebra is $\mathfrak{g} \otimes \mathbb{C}[t]$, with Lie bracket given by $[a \otimes t^m, b \otimes t^n]...
View ArticleDoes the canonical element associated to a finite dimensional $\mathbb{C}^*...
Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let$B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A})...
View ArticleSmallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly...
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent...
View ArticleAre all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact...
Let us consider the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$.Under what assumptions about $\mathfrak{g}$, does the enveloping algebra generate a locally compact...
View ArticleDoes there exist a nontrivial triangular weak Hopf algebra?
Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair($H,\mathcal{R}$) where $H$ is a WHA and$\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(1)$...
View ArticleIs a NC sphere a (one point) compactification of a NC plane?
Inspired by this question About noncommutative sphere and inspired by the fact that the classical sphere is the one point compactificatiin of $\mathbb{R}^2$ we ask the question below:Is the non...
View ArticleWhat are the norms of the generators of the standard Podleś sphere?
Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations\begin{equation*}\begin{split}&a=a^*,~...
View ArticleHall algebra of constructible functions of affine quiver?
I have read in "Quiver Representations and Quiver Varieties" by Kirillov that Hall algebra of constructible functions are defined only for Dynkin quivers because they are of finite type. So is there...
View ArticleCoproduct on $U_q(sl_2)$
Recall that $U_q(sl_2)$ is the quotient of the free associative $\mathbb{Q}(q)$-algebra on generators $E$, $F$, $K^{\pm 1}$ such that $KE = q^2 EK$, $KF = q^{-2} FK$, and $[E,F] = \frac{K - K^{-1}}{q -...
View ArticleA question about q-binomials at roots of unity
I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization".Here is...
View ArticleWhat is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right]...
I have a question about the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. It comes from section $9.3$ on page $300$ of this bookIn Section 9.1, the authors define the...
View ArticleDoes a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?
Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:\begin{align*}& ac = e^{-h}ca, \quad bd =...
View ArticleTsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free
I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to...
View ArticleQuantum Hilbert's fifth problem
Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case.The definition of a quantum...
View ArticleSemisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple...
View ArticleCohomology for quantum groups
I'm interested in quantum groups for two perspectives:Compact quantum groups in the sense of Woronowicz.Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
View Article