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Questions tagged [qa.quantum-algebra]

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

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Finiteness 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 ...
Sebastien Palcoux's user avatar
1 vote
0 answers
125 views

Tangle 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 2-...
Trebor's user avatar
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5 votes
1 answer
179 views

Semisimplicity 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 ...
AffSch's user avatar
  • 61
3 votes
0 answers
267 views

Cohomology 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 &...
user82261's user avatar
  • 357
6 votes
0 answers
349 views

Quantum 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 ...
Sebastien Palcoux's user avatar
7 votes
2 answers
201 views

Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra always have finite order?

Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let $B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
Zhiyuan Wang's user avatar
1 vote
1 answer
64 views

What is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right] _{\varepsilon _i}$ in the restricted specialization in QUE algebras?

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 book In Section 9.1, the authors define ...
fusheng's user avatar
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21 votes
3 answers
808 views

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 ...
Estwald's user avatar
  • 1,391
4 votes
0 answers
183 views

Does 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 = e^{-h}...
yohei ohta's user avatar
2 votes
1 answer
78 views

Does 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(...
Zhiyuan Wang's user avatar
4 votes
1 answer
275 views

What 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^*,~ ...
Zhaoting Wei's user avatar
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2 votes
0 answers
132 views

A 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&...
fusheng's user avatar
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1 vote
0 answers
55 views

Coproduct 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 -...
user536852's user avatar
0 votes
1 answer
144 views

Is 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 ...
Ali Taghavi's user avatar
4 votes
0 answers
82 views

Is the Drinfeld element of a semisimple quasitriangular Hopf algebra invariant under the Drinfeld twist?

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 as $$...
Zhiyuan Wang's user avatar
2 votes
0 answers
45 views

Hall 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 ...
user236626's user avatar
6 votes
2 answers
198 views

Proof that every commutative locally compact quantum group arises from a locally compact group

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 ...
szantag's user avatar
  • 143
10 votes
1 answer
380 views

Braidings on Temperley-Lieb Category

Let $k$ be a field, and let $q\in k^{\times}$. We can then consider the Temperley-Lieb category $TL(q)$. The objects of $TL(q)$ are the non-negative integers, and morphisms are roughly isotopy classes ...
JeCl's user avatar
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13 votes
1 answer
598 views

Is algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a Koszul algebra?

Question: Consider quadratic algebra with four generators $a,b,c,d,d$ and three relations $ac=ca,bd = db, ad-da = cb - bc$ . Is it a Koszul algebra ? (i.e. Koszul complex is resolution of ground field ...
Alexander Chervov's user avatar
3 votes
0 answers
74 views

Are all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact quantum groups?

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 ...
szantag's user avatar
  • 143
3 votes
1 answer
167 views

quantum invariants, ribbon Tannakian duality and classification of ribbon Hopf algebras

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 ...
Léo S.'s user avatar
  • 213
4 votes
0 answers
295 views

Is 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 anisotropic category (namely, the core of ${\mathcal C}$) using finite groups (via the ...
Sebastien Palcoux's user avatar
4 votes
1 answer
246 views

Is 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 $\...
Sebastien Palcoux's user avatar
6 votes
1 answer
231 views

Does Manin's construction of non-commutative endomorphism algebra $\mathrm{End}(A)$ produce Koszul algebra, if $A$ is Koszul?

$\newcommand{\dual}{\mathrm{dual}}\DeclareMathOperator\End{End}\DeclareMathOperator\Fun{Fun}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\GL{GL}$Around 1986–7 Yu.Manin proposed natural and ...
Alexander Chervov's user avatar
4 votes
0 answers
68 views

Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"

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 ...
Zhiyuan Wang's user avatar
10 votes
1 answer
477 views

Where does the definition of ($\infty$-)groupoid cardinality come from?

The cardinality of a finite set $X$ is the number of its elements. Once you know that, you would define the groupoid cardinality of a $\infty$-groupoid $X$ as the quantity $$\lvert X\rvert := \sum_{[x]...
Matthew Niemiro's user avatar
2 votes
0 answers
48 views

Proof 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]...
Saima Samchuck-Schnarch's user avatar
1 vote
0 answers
70 views

Affiliating the whole algebra of 'coordinates' with a locally compact quantum group

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 ($A_q$) ...
szantag's user avatar
  • 143
4 votes
1 answer
161 views

Existence of an element in a Hopf algebra that satisfies a 'flip' property

Consider a Hopf algebra $H$ where $S$ is the antipode and $\Delta$ is the coproduct. Given $a \in H$, I want to know if there always exists an element $b \in H$ satisfying the 'flip' property: $$\...
pyroscepter's user avatar
3 votes
0 answers
106 views

Monoidal 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 ...
Damalone's user avatar
  • 151
4 votes
0 answers
227 views

Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?

In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension". From [DLN, Theorem II (iii)], where the ...
Sebastien Palcoux's user avatar
5 votes
1 answer
156 views

Explicit correspondence between classical double and quantum double

Proposition 12.3 of Etingof and Schiffmann's "Lectures on Quantum Groups" states the following claim. Proposition 12.3. Let $H$ be a quantized enveloping algebra and let $\mathfrak{g}$ be ...
yohei ohta's user avatar
3 votes
0 answers
107 views

How to get $U(N)_k$ Kac-Moody modules and characters from $N \cdot k$ Dirac Fermions using $U(N \cdot k)_1 / SU(k)_N$?

It is known that the $U(N)_k$ Kac-Moody algebra can be written as the coset $U(N)_k = U(N \cdot k)_1 / SU(k)_N$. (This fact is related to the level-rank duality of $U(N)_k \leftrightarrow U(k)_N$.) A ...
Joe's user avatar
  • 545
4 votes
0 answers
166 views

Coloured Jones polynomial at 4th root of unity and Arf invariant

Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
hopftype's user avatar
2 votes
1 answer
183 views

Poisson quantization vs quantization in atomic physics

Is it possible to interpret quantization in atomic physics ( e.g. the quantization condition in hydrogen atom stated as exponential decay of wave functions at infinity and analogously for n-electron ...
0x11111's user avatar
  • 593
4 votes
0 answers
326 views

Are 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 $(\FPdim(b_i)...
Sebastien Palcoux's user avatar
3 votes
0 answers
100 views

Isomorphic objects have the same dimension (pivotal categories)

I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e., $$ \mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...
NoetherNerd's user avatar
1 vote
0 answers
156 views

Is anything known about the derivative of the quantum dilogarithm?

Faddeev's noncompact quantum dilogarithm is the function defined by $$ \Phi_{\mathsf b}(z) = \exp \int_{\mathbb{R} + i\varepsilon} \frac{ e^{-2i zw} }{ 4 \sinh(w \mathsf b ) \sinh(w/\...
Calvin McPhail-Snyder's user avatar
3 votes
0 answers
97 views

What algebras generate polynomial count varieties as their representations spaces ? Is it preserved by the Koszul duality, Manin's endomorphisms?

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 ...
Alexander Chervov's user avatar
5 votes
1 answer
349 views

Modularity of the Drinfeld center of the category of G-graded vector spaces

Background: Let $G$ be a finite group, and $\mathrm{Vect}_G$ be the category of finite dimensional $G$-graded vector spaces over some algebraically closed field $k$ of char 0. It is well-known that $\...
Xiaomeng Xu's user avatar
2 votes
1 answer
312 views

Reasons about the difference between twisted affine algebras of $A_{2l}$ and other types

I am reading about Kac–Moody algebras. I have a question about the construction of twisted affine algebras. Let $\sigma$ be a graph automorphism of the simple Lie algebra $\mathfrak{g}$. When $\...
fusheng's user avatar
  • 137
1 vote
0 answers
50 views

Simple highest weight modules of quantum affine algebras

Let $U_q(\hat{\mathfrak g})$ be a quantum affine algebra, and let $L$ be an integrable simple highest weight module of $U_q(\hat{\mathfrak g})$. In [Lu], Lusztig proved that the limit of $L$ when $q\...
qweqwe's user avatar
  • 11
1 vote
0 answers
72 views

Root systems of Weyl groupoids

I am working with the notion of Weyl Groupoids as introduced in "A generalization of Coxeter groups, root systems, and Matsumoto’s theorem" by Heckenberger and Yamane. The authors generalize ...
Tim's user avatar
  • 11
6 votes
1 answer
224 views

Does $U_q (\mathfrak{sl}_2)$ have a universal $R$-matrix?

Consider the standard quantum group $U_q (\mathfrak{sl}_2)$ over the field $\mathbb{C}(q)$ of rational functions (or over $\mathbb{C}$ if $q \in \mathbb{C}$ is not a root of unity), with the usual ...
Minkowski's user avatar
  • 601
5 votes
1 answer
506 views

Generalized Wigner 3-j symbol and Legendre functions

Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc} n & m & h\\ 0 & 0 & 0 \end{array}\right)^{2}\tag{...
User's user avatar
  • 219
87 votes
11 answers
14k views

What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
Julio César Salazar García's user avatar
55 votes
2 answers
11k views

What is quantum algebra?

This might be a very naive question. But what is quantum algebra, really? Wikipedia defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a ...
Najib Idrissi's user avatar
5 votes
0 answers
128 views

Fusion categories with $\mathrm{PSU}(2)_k$ fusion rules

Let $R_k$ be a fusion ring with $\mathrm{SU}(2)_k$ fusion rules (or equivalently $A_{k+1}$ fusion rules). All fusion categories with such fusion rules have been classified by Frohlich and Kerler in ...
Gert's user avatar
  • 293
7 votes
0 answers
151 views

How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?

Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity. Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...
matha's user avatar
  • 193
8 votes
1 answer
390 views

What is the ring structure on Lusztig's integral form of quantum $\mathfrak{sl}(2)$?

Consider the quantum group $U_q(\mathfrak{sl}_2)$, with generators $E,F,K$ such that $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$. Write $[n]=\frac{q^n-q^{-n}}{q-q^{-1}}$, and $[n]!=[n][n-1]\dotsm[1]$. In ...
Alvaro Martinez's user avatar

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