Questions tagged [qa.quantum-algebra]

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

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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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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
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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
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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
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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
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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
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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 ...
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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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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
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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 $(\FPdim(b_i)...
Sebastien Palcoux's user avatar
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Where does the definition of ($\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
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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 $\...
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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
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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
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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 ...
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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
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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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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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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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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
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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
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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
410 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
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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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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
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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
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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
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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 ...
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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
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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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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
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$U_q(\mathfrak{g})$ 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
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2 answers
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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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The dual of elements $E$, $F$, and $H$ of $U_h(\mathfrak{sl}_2)$ corresponds to which element of $F_h(\mathrm{SL}_2)$ by isomorphism?

$\newcommand{\sl}{\mathfrak{sl}}\DeclareMathOperator\SL{SL}$Let $U_h(\sl_2)$ be the quantized universal enveloping algebra of $\sl_2(\mathbb{C})$ and $F_h(\SL_2)$ be the quantized function algebra of $...
yohei ohta's user avatar
3 votes
0 answers
121 views

Transferred $L_\infty$-structure from Hochschild dgLA

Let $D_{poly}$ be the differential graded Lie algebra (dgLA) of differentiable Hochschild cochains on a manifold $\mathscr M$, endowed with the usual Gerstenhaber bracket $[-,-]_G$ and Hochschild ...
thingsthatmighthavebeen's user avatar
4 votes
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Hopf algebra and coideal question

Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put $$B^+:= B\cap \ker(\epsilon).$$ It can be shown that $B^+$ is a two-...
Andromeda's user avatar
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Automorphism group of the quantum Weyl field

Let $\mathsf{k}$ be a field with zero characteristic, and $q \in \mathsf{k}$ a non-zero elemento which is not a root of unit. The quantum plane $\mathsf{k}_q[x,y]$ is the algebra given by generators $...
jg1896's user avatar
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Relation between "homotopical" and "representation-theoretic" categorifications

This might be a bit of a soft question, and I apologize in advance for this. Here it is: What is the relationship between the "homotopical" categorification (e.g. we consider every category ...
Grisha Taroyan's user avatar
6 votes
1 answer
241 views

Quantum exterior algebra

In Generalisation of the quantum exterior algebra the quantum exterior algebra is discussed: $$ K\langle x_1,\dotsc x_n\rangle/(x_i^2,x_i x_j + q_{i,j}x_j x_i), $$ with nonzero field elements $q_{i,j}...
László Szabados's user avatar
5 votes
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189 views

parameter of a quantum group

I am currently learning about quantum groups, and I got a question about how two different ways of thinking the $q$-parameter of quantum groups are related to each other. Here by a quantum group, I ...
Ji Woong Park's user avatar
2 votes
1 answer
147 views

Question on tensor product of von Neumann algebras and subfactors

Let $M_1$ and $M_2$ be von Neumann algebras acting on Hilbert spaces $H_1,H_2$ and consider $M=M_1\overline\otimes M_2$ acting on $H_1\otimes H_2$. Let $K$ be an $M$-invariant subspace (so that $P_K\...
Lau's user avatar
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How to make sense of $\mathrm{Mat}_q(n \times n)$? Are there notions of quantum vector space, quantum linear algebra, etc?

Given some algebra $\mathcal{A}$ and $q \in \mathbb{C}$, we say that a matrix $M \in \mathrm{Mat}(n \times n ; \mathcal{A})$ is a quantum matrix in $\mathrm{Mat}_q(n \times n)$ iff the following ...
Joe's user avatar
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Is there a classical version of Yetter-Drinfeld modules?

One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$. If we think ...
Antoine Labelle's user avatar
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114 views

Points and algebraic geometry on the quantum plane

The "quantum plane" is the "space" of the algebra $A=\Bbbk\langle X,Y\rangle/(YX-qXY)$, for a scalar $q$ (e.g. $\Bbbk=\mathbb C(q)$). I would like to know how much algebraic ...
grok's user avatar
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5 votes
1 answer
219 views

Completely isometric coaction of discrete quantum group is multiplicative?

Let $\mathbb{G}$ be a compact quantum group (in the sense of Woronowicz) with discrete dual $\widehat{\mathbb{G}}$ which we view as a von Neumann algebraic locally compact quantum group (in the sense ...
J. De Ro's user avatar
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2 votes
0 answers
80 views

DHR superselection and DR reconstruction in low spacetime dimensions

Given a completely rational net on $\mathbb{R}$, the Doplicher-Haag-Roberts (DHR) category is a modular fusion category (MFC) identical to that associated with the corresponding vertex operator ...
Ying's user avatar
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3 votes
1 answer
111 views

Unitary in adjointable operators associated with equivariant Hilbert module

Consider the following fragment from the article "Tannaka–Krein duality for compact quantum homogeneous spaces. I. General theory" by De Commer and Yamashita: How exactly is $\mathcal{E}\...
Andromeda's user avatar
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4 votes
1 answer
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Reference request: decomposability of $\mathbb{G}$-Hilbert modules

Let $\mathbb{G}$ be a compact quantum group, $B$ be a $C^*$-algebra together with a right action $$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $*$-homomorphism satisfying $(\beta \...
J. De Ro's user avatar
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0 answers
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Is the Frobenius property invariant by Morita equivalence?

Kaplansky's sixth conjecture [Ka75] states that the dimension of a semisimple finite dimensional Hopf algebra over $\mathbb{C}$ is divisible by the dimension of its irreducible complex representations....
Sebastien Palcoux's user avatar
2 votes
0 answers
103 views

Is there a non-pointed simple integral modular fusion category?

The weakly group-theoretical conjecture (supporting a negative answer to [ENO11, Question 2]) states as follows: Statement 1: Every integral fusion category is weakly group-theoretical. We wonder ...
Sebastien Palcoux's user avatar

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