# Tag Info

### 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 (...
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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 ...
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### 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 "...

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### 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 ...
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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 ...
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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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### 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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### 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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### 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 ...

### 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 ...

### 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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### 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, ...
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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). ...
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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 ...
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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, ...
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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 ...
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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  ...
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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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### 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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### 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 ...
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