When does Morita equivalence between two Hopf-von Neumann algebras imply also...
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...
View ArticleAn alternative Cauchy theorem on Hopf algebras
Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.There already exists a generalization of Cauchy theorem using exponent, see [KSZ06].We are interesting in an alternative...
View ArticleEstimating ground state energy of $n$-qubit $2$-local Hamiltonian $H$ with...
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 like...
View ArticleModularity 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...
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Reasons about the difference between twisted affine algebras of $A_{2l}$ and...
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...
View ArticleWhat 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...
View ArticleWhere 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$$\lvert X\rvert := \sum_{[x]...
View ArticleRings or algebras with many nilpotent elements and efficient computation
Crossposted from quantum.SEwhere comment appears to suggest that solving modulo 2 mightbe possible.Searching the web for '"quantum computer" nilpotent'returns many results, so maybe the question is...
View ArticleAre there infinitely many simple integral fusion rings of rank $4$?
$\DeclareMathOperator\ch{ch}$$\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...
View ArticleExplicit 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...
View ArticleRoot 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 the...
View ArticlePoisson 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...
View ArticleIs anything known about the derivative of the quantum dilogarithm?
Faddeev's noncompact quantum dilogarithm is the function defined by$$ \Phi_{\mathsf b}(z) = \exp \int_{\mathbb{R} + i\varepsilon} \frac{ e^{-2i zw} }{ 4 \sinh(w \mathsf b ) \sinh(w/\mathsf b) }...
View ArticleExistence of an element in a Hopf algebra that satisfies a 'flip' property
Consider a Hopf algebra $H$ where $S$ is the antipode and $\Delta$ is the coproduct. Given $a \in H$, I want to know if there always exists an element $b \in H$ satisfying the 'flip' property: $$\Delta...
View ArticleColoured 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...
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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}) =...
View ArticleA 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...
View ArticleIs the rank of an integral MTC an upper bound for the prime factors of its...
In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension".From [DLN, Theorem II (iii)], where the modular data is normalized, we get (see...
View ArticleExamples 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 algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a...
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...
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