53 votes
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What is quantum algebra?

Quantum algebra is an umbrella term used to describe a number of different mathematical ideas, all of which are linked back to the original realisation that in quantum physics, one finds ...
Jan Grabowski's user avatar
26 votes
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What are the matrices preserving the $\ell^1$-norm?

As pointed out by YCor in the comments, the following theorem is true: Theorem 1 Let $p \in [1,\infty] \setminus \{2\}$. If a matrix $A \in \mathbb{R}^{n \times n}$ is an isometry on $\mathbb{R}^n$ ...
Jochen Glueck's user avatar
24 votes
Accepted

Why are quantum groups so called?

Typically in math "quantum X" means a deformation of "X" which is in some sense "less commutative." So quantum groups should be deformations of groups which are "less commutative." Interpreting this ...
Noah Snyder's user avatar
  • 27.8k
23 votes
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Is there any published physics article where $q$-mathematics is applied?

There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-...
Konstantinos Kanakoglou's user avatar
20 votes

Quantum corrections to geometry

Physicist chiming in; "quantum corrections of a geometry" represents the vague idea that gravity should be quantum. Classically, gravity is a geometric theory: "reality" is modelled as a (Lorentzian) ...
AccidentalFourierTransform's user avatar
17 votes
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Where does the definition of ($\infty$-)groupoid cardinality come from?

I'll restrict to $\pi$-finite spaces (where the definition is guaranteed to make sense). Then homotopy cardinality is multiplicative in fiber sequences: if $E \to B$ is a fibration with connected base ...
Phil Tosteson's user avatar
15 votes
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$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?

Yes, there's a $q$-analogue. See this paper of Yoshida from 1992, where $\sum_{n \ge 0} \frac{| \mathrm{Hom}(G,\mathrm{GL}(n,\mathbf{F}_q)) | }{ |\mathrm{GL}(n,\mathbf{F}_q)| } z^n$ is expressed in ...
Ofir Gorodetsky's user avatar
15 votes
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Conjectures of Peter Scholze about q-de Rham complex: examples

Good question! I wished I understood what this conjecture really means, concretely. First, I should say that in my paper with Bhatt on prismatic cohomology, much of the content of these conjectures ...
Peter Scholze's user avatar
15 votes
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Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?

There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do ...
Konstantinos Kanakoglou's user avatar
14 votes
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Classification of unitary modular tensor categories (UMTCs)

Recently, Mignard and Schaunberg found a counter example to the conjecture (https://arxiv.org/abs/1708.02796). Counting $(S,T)$ pairs is still just a shorthand way of counting monoidal equivalence ...
Matthew Titsworth's user avatar
13 votes

Where does the name "R-matrix" come from?

This answer refers to what is probably the first appearance of an $R$-matrix in the context of quantum mechanical scattering theory. Quite possibly the later appearances in the context of the inverse ...
Carlo Beenakker's user avatar
13 votes

Can one define quantized universal enveloping algebras in a basis-free way?

For complex simple $\mathfrak g$, Drinfeld (1986, p. 807) already characterized his $\mathrm U_h\mathfrak g$ as the unique (up to equivalence and change of parameter) deformation of $\mathrm U\...
Francois Ziegler's user avatar
13 votes

What are the matrices preserving the $\ell^1$-norm?

There's a very simple approach in finite dimension. Let $G$ be the linear isometry group of $(\mathbf{R}^n,\|\cdot\|_p)$, $1\le p\le\infty$. Let $W$ be the group of signed permutations. First, since $...
YCor's user avatar
  • 60k
13 votes

Axiomatic definition of quantum groups

I would have liked to write this as a comment, but with my points tally I can not. So writing this as an answer. In quantum groups, we are probably at a stage group theory was, say in the first half ...
akp's user avatar
  • 311
12 votes

What is quantum algebra?

I think that a modern realistic perception of the term "quantum algebra" has to be understood in its historical context, that is, the algebraic/geometric methods, originating from the study of the ...
Konstantinos Kanakoglou's user avatar
12 votes
Accepted

Hopf dual of the Hopf dual

I am going to give three counterexamples to your first question. (The third counterexample is courtesy of @Adrien, who did most of the job.) While none of them leads to a full answer of your second ...
darij grinberg's user avatar
12 votes
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Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

That the only monoidal deformations of the category of representations of $U(\mathfrak{g})$ is the category of representations of $U_q(\mathfrak{g})$ is known in Type A from Kazhdan-Wenzl (Adv. Soviet ...
Noah Snyder's user avatar
  • 27.8k
12 votes
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Is a Hopf algebra a group object of some category?

Not with their definition, where they assume the underlying category to be cartesian. You can define a notion of "Hopf object" in arbitrary symmetric monoidal categories, where you also need ...
Adrien's user avatar
  • 8,232
12 votes
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Hopf algebra with a non-invertible antipode

Theorem of Takeuchi (in Free Hopf algebras generated by coalgebras, 1971) asserts that free Hopf algebra $H(C)$ over a coalgebra $C$ has injective antipode, and it is bijective precisely (at least ...
Denis T's user avatar
  • 4,309
11 votes
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Where does the name "R-matrix" come from?

(See update at the bottom) I could not resist: in the meanwhile I have searched the literature further. I think the following is an interesting addition to my best guess from the OP, which is why I ...
Jules Lamers's user avatar
  • 1,683
11 votes

Is there any published physics article where $q$-mathematics is applied?

As another example of the second category in Kostantinos Kanakoglou's answer I think it is fair to mention quantum-integrable systems: this topic in physics was pivotal in the historical development ...
Jules Lamers's user avatar
  • 1,683
11 votes

Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$

This is a very interesting question. I have also made some search but i have not found this result explicitly mentioned somewhere in the literature. However, i remember i have heard such a claim in ...
Konstantinos Kanakoglou's user avatar
11 votes
Accepted

Why is Planar algebras I (by Vaughan Jones) not published?

This paper is now published in New Zealand Journal of Mathematics Vol. 52 (2021). https://doi.org/10.53733/172 pdf file
Keshab Bakshi's user avatar
11 votes

Quantum corrections to geometry

I presume this refers to quantum corrections to the metric tensor. These are expected to be important in general relativity when the curvature of space-time approaches the Planck scale $\sqrt{G\hbar/c^...
Carlo Beenakker's user avatar
11 votes
Accepted

Is there a strongly noncommutative fusion category?

Consider the symmetric group group $G = S_3$ of order $6$. Then $\mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1)) \cong \mathbb Z/6\mathbb Z$. Choose a generator $\omega \in \mathrm{H}^3_{\mathrm{gp}}(G;\...
Theo Johnson-Freyd's user avatar
10 votes
Accepted

Name for the action of a bialgebra on an algebra

According to nLab, such an action is called a Hopf action and your data specify a left $B$-module algebra. Such a structure is also referred to in the literature as an algebra in the category (of left ...
Konstantinos Kanakoglou's user avatar
10 votes

$(\infty,1)$ 2d TFTs

The answer to your question is known when $\mathcal{S}$ is a symmetric monoidal $\infty$-groupoid, by work of Galatius-Madsen-Tillmann-Weiss. In other words: we understand invertible $2$-dimensional ...
Jan Steinebrunner's user avatar
10 votes

Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

$\newcommand{\g}{\mathfrak g}$ I think the statement Scott Carnahan was refeering to in his answer concerns in fact formal deformations of representations of $\g$, i.e. deformations over the ring $\...
Adrien's user avatar
  • 8,232
10 votes
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q-difference equations and quantum mechanics

There exist applications of q-calculus to physics, but there is no direct relation to quantum mechanics. You can find an overview of some of these applications in q-Calculus and physics (paywall). ...
Carlo Beenakker's user avatar
10 votes
Accepted

Drinfeld center of a braided category

No, the functor $\mathcal C \to \mathcal Z(\mathcal C)$ is not essentially surjective in general. For example, in the case you have in mind, $\mathcal C = Rep_q(G)$ (say $G$ a semisimple algebraic ...
Sam Gunningham's user avatar

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