55
votes

### Intuition behind the definition of quantum groups

In algebraic combinatorics, there is an important concept of a "$q$-analogue". Surprisingly often when you have a counting problem with a good integer answer, you realize that it can be refined to a (...

50
votes

Accepted

### 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 ...

33
votes

### Intuition behind the definition of quantum groups

Here is an answer to question (1). I recommend that you split of question (2) as a separate question.
Define the quantum plane to be the "spectrum" of the noncommutative ring $\mathbb K\langle x,y\...

25
votes

Accepted

### 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 $\mathbb{R}^n$ ...

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 ...

23
votes

Accepted

### 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 $q$-...

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) ...

15
votes

Accepted

### 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 ...

14
votes

Accepted

### $q$-(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 ...

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\...

13
votes

Accepted

### 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 ...

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 $...

13
votes

Accepted

### 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 ...

12
votes

### Intuition behind the definition of quantum groups

To follow up on Greg's answer, here is a simple explanation as to how quantum groups can be thought of as $q$-analogs of semisimple Lie algebras. Let's just take $ sl_2 $ for simplicity.
Recall that ...

12
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 ...

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 ...

12
votes

Accepted

### 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 ...

12
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 ...

12
votes

Accepted

### 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 ...

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 ...

11
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 ...

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 ...

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

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^...

11
votes

Accepted

### 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 ...

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;\...

10
votes

Accepted

### Yang–Baxter explanation

One very nice family of examples is the one of set-theoretical solutions.
In the paper
Drinfelʹd, V. G. On some unsolved problems in quantum group theory. Quantum groups (Leningrad, 1990), 1--8, ...

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 ...

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 ...

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 $\...

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