Smallest 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 ArticleTangle hypothesis and ribbon category
The tangle hypothesis, when specialized to ordinary framed tangles, says that the framed tangles form the free braided category with all duals (i.e. considered as a 3-category, all the 1- and...
View ArticleFiniteness of the number of Hopf subalgebras
Let $H$ be a finite-dimensional Hopf algebra over the complex field.Question: Does $ H $ have a finite number of Hopf subalgebras?In the case where $ H $ is semisimple, the answer is yes. According to...
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Frobenius identity on finite-dimensional Hopf algebras
Let $(H,m_H,e_H,\Delta_H,\epsilon_H)$ be a finite-dimensional Hopf algebra over $\mathbb{C}$, where $m_H$ denotes multiplication, $e_H$ is the unit map, $\Delta_H$ is the comultiplication, and...
View ArticleAre there infinitely many simple integral fusion rings of rank $4$?
$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list...
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q->0 limits of q-special functions and q-identities - interesting examples...
There are many beautiful facts in "q-analogs world": q-special functions (e.g. basic hypergeometric series , q-identities (e.g. quantum dilogarithm related). Typically q->1 we return to classical...
View ArticleAre there q-analogs of GKZ (Gelfand-Kapranov-Zelevinsky) hypergeometric...
GKZ (Gelfand-Kapranov-Zelevinsky) hypergeometric differential equations are generalisations of standard hypergeometric equations/functions and important for example for mirror symmetry. It is system of...
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