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Questions tagged [qa.quantum-algebra]

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

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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 ...
Lagrenge's user avatar
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6 votes
2 answers
137 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 $B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
Lagrenge's user avatar
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2 votes
0 answers
31 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
67 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
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2 answers
165 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
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268 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
3 votes
0 answers
88 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
3 votes
1 answer
139 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
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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 $$...
Lagrenge's user avatar
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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
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1 answer
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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
10 votes
1 answer
351 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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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
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97 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
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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
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13 votes
1 answer
581 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
20 votes
3 answers
700 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
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A question about the existence of rational functions

I am reading a paper Representations of shifted quantum affine algebras. I have a question about the existence of a rational function about the remark $4.4$ I'll briefly describe the problem. We let $...
fusheng's user avatar
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0 answers
223 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
3 votes
0 answers
96 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
4 votes
0 answers
146 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
4 votes
1 answer
146 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
1 vote
0 answers
146 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
2 votes
1 answer
170 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
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1 vote
0 answers
66 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
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4 votes
1 answer
142 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
4 votes
0 answers
320 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
10 votes
1 answer
444 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
1 answer
274 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
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0 votes
0 answers
29 views

Estimating ground state energy of $n$-qubit $2$-local Hamiltonian $H$ with known coefficients

Suppose we have an $n$-qubit $2$-local Hamiltonian $H$ with known coefficients. The eigenvalues of $H$ lie in $[0,1)$ and can all be written exactly with $[2 \log_2n]$ bits of precision. You would ...
QiFeng233's user avatar
5 votes
1 answer
279 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
4 votes
0 answers
51 views

When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?

Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...
szantag's user avatar
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2 votes
0 answers
111 views

Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?

Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})...
yohei ohta's user avatar
5 votes
0 answers
116 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
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6 votes
1 answer
185 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
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6 votes
0 answers
127 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
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2 votes
0 answers
25 views

Examples of (weak-)bialgebras/Hopf algebras with a finite dimensional unitary representation and corepresentation and polynomial growth rate

I need examples (the more the better, even better if there is a systematic way of construction) of (weak-)bialgebras or (weak-)Hopf algebras $H$ with a finite dimensional representation $\rho$ and a ...
Lagrenge's user avatar
  • 575
3 votes
1 answer
172 views

A filtration on Drinfeld-Jimbo quantum enveloping algebras

For the universal enveloping algebra $U(\frak{g})$ of a Lie algebra $\frak{g}$, one can define in a natural way an increasing $\mathbb{N}_{0}$-filtration. By the Poincaré-Birkhoff–Witt theorem, the ...
Lorenzo Del Vecchiopontopolos's user avatar
7 votes
0 answers
290 views

Does the pentagon axiom force the associativity constraint to be a natural isomorphism?

Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...
Sebastien Palcoux's user avatar
5 votes
1 answer
459 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
0 votes
0 answers
102 views

A variant of quantum harmonic oscillators

We have the following variant of harmonic oscillators. $$ \left\{ \begin{array}{**lr**} T = a + a^\dagger\\ a | n \rangle = \sqrt{[n]} |n-1 \rangle \\ a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
Lili Si's user avatar
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2 votes
0 answers
96 views

Questions about proof that all indecomposable module categories over $\operatorname{Rep}(G)$ are equivalent to $\operatorname{Rep}^1(H,\omega)$

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\End{End}$In Ostrik - Module categories, weak Hopf algebras and modular invariants, it is ...
shin chan's user avatar
  • 301
3 votes
1 answer
178 views

Quantum group associated to a reductive group

In most of the classical references about quantum groups, these objects are defined as a one-parameter deformation of the universal enveloping algebra. However, I have read in several papers that it ...
jeykey's user avatar
  • 31
4 votes
1 answer
163 views

Drinfeld-Jimbo quantum groups for $q=0$

In the Wikipedia page of Drinfeld--Jimbo quantum groups the values of $q=0,1$ are excluded so as to avoid dividing by zero. The $q=1$ case is discussed in this old question. What about the $q=0$ case? ...
Jake Wetlock's user avatar
  • 1,154
2 votes
0 answers
100 views

Morphism of discrete quantum groups

In the paper Kazhdan's Property T for Discrete Quantum Groups , we read the following fragment: First, note that I think there is a typo and that codomain and domain of the dual maps have to be ...
Andromeda's user avatar
  • 169
1 vote
1 answer
79 views

Show that if $V\in M(B_0(H)\otimes A)$, then $V(B(H)\otimes 1)V^*\not\subseteq B(H)\otimes A$ where $A$ is a specific unital $C^*$-algebra

Let $\mathbb{G}$ be a compact (quantum group) with function algebra $(C(\mathbb{G}), \Delta)$ and Haar state $\varphi_{\mathbb{G}}$. Consider the associated GNS-representation $\pi_{\mathbb{G}}: C(\...
Andromeda's user avatar
  • 169
7 votes
1 answer
365 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
0 votes
0 answers
96 views

Some version of non-commutative Wick formula

Let $V$ be a vertex algebra. The traditional non-commutative Wick formula is a tool to calculate term like $[a_\lambda:bc:]$. However, I need to calculate terms of the form $[:ab:_\lambda c]$. I found ...
Estwald's user avatar
  • 1,341
8 votes
0 answers
227 views

$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$. Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
Pulcinella's user avatar
  • 5,565
3 votes
2 answers
212 views

Simple modular tensor category and zero entries in its S-matrix

Question 1: Is there a simple modular fusion category with a zero entry in its S-matrix? (or equivalently, with a fusion matrix of zero determinant?) Yes, by this answer below providing the example $\...
Sebastien Palcoux's user avatar

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