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 (...
49 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 ...
34 votes

Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The answer of Jørgen Ellegaard Andersen only concerns the case of when the gauge group is $SU(n)$. I will argue that all the ingredients for the equivalence between the two approaches (namely "...
32 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\...
24 votes
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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 ...
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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 $q$-...
17 votes
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Tips to organize a successful math workshop

The Lorentz Center has some advice that you might find useful, I have organized several workshops there and followed a route similar to the one you describe. Tentative answers to your specific ...
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 ...
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\...
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
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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
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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 ...
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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 ...
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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 ...
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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

Hopf structure on the universal enveloping of a super Lie algebra

This is true. In other language, if I understand rightly, a super Lie algebra is just a graded Lie algebra with grading over {0,1} (even and odd), with the standard sign conventions as in algebraic ...
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10 votes
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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
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Cartier-Kostant-Milnor-Moore theorem

When $k$ fails to be algebraically closed the theorem is false but the discrepancy can be understood in terms of Galois descent and so in principle understood in terms of Galois cohomology. Suppose $...
10 votes
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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

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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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). ...
9 votes
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Computing in quantum groups

There is the package QuaGroup by de Graaf for both GAP and Magma: see QuaGroup. I've used it in both systems and found it to be extremely helpful. Since there's a GAP package, you also have the ...
9 votes
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Algebra in a category

What you are talking about is the notion of monoid in a monoidal category. To show $A$ is a monoid ('algebra'), you need to construct a multiplication map $\mu: A \times A \to A$, that is associative, ...
9 votes
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An inner product approach to Hopf algebras

This doesn't directly answer your question concerning Hopf structures on $\mathbb{C}^n$, but a particularly well-studied class of Hopf algebras for which the product is the adjoint of the coproduct ...
9 votes
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What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?

Your two equations are equivalent, and are both versions of the quantum YBE. (The question from the comments does a good job of answering your classical versus quantum question.) Write the first as $$ ...
8 votes
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Brauer-Picard for a fusion category coming from a quantum group

As far as I know, no one has written this up, but I think you should be able to find the Brauer-Picard groupoid for quantum groups at roots of unity by the following techniques. Now that I've written ...
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8 votes
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Understanding "Decategorified" symplectic Khovanov homology

One thing you didn't mention is the manifold in which this calculation happens: the Slodowy slice to a nilpotent of type $(n,n)$. This manifold is the key to everything, since it is a geometric avatar ...
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8 votes
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Is there some relation between cluster algebras and crystal graphs?

Yes, there are many relations between cluster algebras and crystal graphs. I am by no means an expert on these things, but let me mention one connection. Cluster algebras were originally discovered in ...
8 votes

What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases?

I am not an expert (far from), but have done some reading on this myself a while back. So, I will share what I have found. I will summarize a few things below. Though the one of the best resources I ...

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