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 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 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$B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A})...
View ArticleDoes there exist a nontrivial triangular weak Hopf algebra?
Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair($H,\mathcal{R}$) where $H$ is a WHA and$\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(1)$...
View 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...
View ArticleRigged Hilbert space and function of ultra-rapid decay
Consider the Hilbert space $\ell^2(\mathbb{Z})$ and unbounded operators $u$ and $v$ defined as $u: f_k \to f_{k+1}$ and $v: f_k \to q^k f_k$ for some fixed $0<q<1$. Together with $u^{-1}$ and...
View ArticleMotivation for twisted Yangians?
I am asking this question, hoping to understand the motivation behind the construction of the so-called twisted Yangians. This will be a long question, so my thanks in advance for any comment and/or...
View ArticleIntuition behind the definition of quantum groups
Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to...
View ArticleHausdorff dimension and von Neumann dimension
Non-integral dimensions arise in two key areas:fractal geometry: This includes the well-known Hausdorff dimension of fractals,von Neumann algebra: In this context, consider a type ${\rm II_1}$ factor...
View ArticleEquivalent definitions of the simple $\ell$-roots $A_{i,a}$ of quantum affine...
Let $U_q(\widehat{\mathfrak g})$ be the quantum affine algebra over a simple Lie algebra $\mathfrak g$. I am trying to understand and compare the so called simple $\ell$-roots $A_{i,a}$ seen in both...
View ArticleSubalgebra of quantum groups from extended Dynkin diagrams
Given a semisimple Lie algebra, we can construct subalgebras corresponding to removing nodes from the extended Dynkin diagram. For example $\mathfrak{su}_3 \subseteq \mathfrak{g}_2$. I'm interested in...
View ArticleHighest-weight modules for RTT-quantum groups
First of all, my question is motivated by the following phenomenon. Suppose that $Y$ is a quantisation of a Lie bialgebra (say $y$), and suppose that $Y' \subset Y$ be a coideal subalgebra. In the...
View ArticleIs the topological dual of a quantized enveloping algebra a Hopf algebra?
I am looking for a proof of Proposition 10.2 in Etingof and Schiffmann’s Lectures on Quantum Groups. In particular, I would like to understand the proof of the following statement in an elementary...
View ArticleSecond Quantization with Coulomb potential
I am trying to understand how one can perform second quantization in the case of the Hamiltonian of the hydrogen atom, i.e. when the one particle Hamiltonian acquires an external coulomb potential. The...
View ArticleExplicit description of the Poisson–Lie group associated with the Drinfeld...
Let $\mathfrak{b}_+$ be the Borel subalgebra of $\mathfrak{sl}_2$. It is equipped with a Lie bialgebra structure given by$$[H, E] = 2E, \quad \delta(E) = E \otimes H - H \otimes E, \quad \delta(H) =...
View ArticleThe other classical limit of a quantum enveloping algebra?
Let $\mathbb K$ be a field (of characteristic 0, say), $\mathfrak g$ a Lie bialgebra over $\mathbb K$, and $\mathcal U \mathfrak g$ its usual universal enveloping algebra. Then the coalgebra structure...
View ArticleIs every finite quantum group a quantum symmetry group?
This post is basically a quantum extension of Is every finite group a group of “symmetries”?Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra. Frucht's theorem states...
View ArticlePractical Ways to get Skew-Schur Functions
The Schur polynomials satisfy many, many identities and there is a whole book about them.I think the easiest way is with the Vandermonde Determinant.$$s_{3,1,1}(a,b,c) = \frac{\left|\begin{array}{ccc}...
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