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
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
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 ...
16 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
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

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
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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 ...
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
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
  • 1,813
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
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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 ...
Mark Penney's user avatar
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 ...
Peter May's user avatar
  • 30.2k
10 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, ...
Jacob White's user avatar
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 $...
Qiaochu Yuan's user avatar
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 ...
Konstantinos Kanakoglou'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
  • 8,234
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
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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 $$ ...
Calvin McPhail-Snyder's user avatar
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 ...
Jan Grabowski's user avatar
9 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 ...
John Machacek's user avatar
9 votes

DW, state sum models, and fully extended TQFTs

Let me try to answer your questions at least in part. My apologies for references I've missed. For an overview of the ideas without references, you might enjoy Pavel Safranov's talks at the intro ...
Noah Snyder's user avatar
  • 27.8k
9 votes
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Characterizing discrete quantum groups

Also the implication (2) $\Rightarrow$ (1) holds and can be proven as follows. Denote by $\mathcal{C}$ the category of all finite dimensional, nondegenerate $*$-representations of $M$. The morphisms ...
Stefaan Vaes's user avatar
  • 4,011
9 votes
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Low dimensional noncommutative non-cocommutative Hopf algebras

By standard results (in fin dim, over an alg closed field of zero char), all cocommutative HAs are group algebras (for some finite group), all commutative HAs are duals of group HAs (for some finite ...
Konstantinos Kanakoglou's user avatar
9 votes
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In the rep theory of Quantum Double, why does the fusion of 2 "pure fluxes" yield a "pure charge"?

Think of reps of $D(G)$ as $G$-equivariant vector bundles on $G$, where the group acts by conjugation. In this language, the tensor product is push-forward under multiplication $\mu : G\times G \...
Bugs Bunny's user avatar
  • 12.1k
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 ...
Ben Webster's user avatar
  • 43.9k
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 ...
Spencer Leslie's user avatar
8 votes
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Inner automorphisms of Hopf algebras

I am not sure if the following is the kind of answer you are expecting, but take the (left) adjoint action $(ad_l h)\triangleright k=\sum h_1 kS(h_2)$ of a hopf algebra $H$ on itself. (It is known ...
Konstantinos Kanakoglou's user avatar
8 votes

P-adic Volume Conjecture

I don't know how to answer your question, since I don't know about motives or $p$-adic regulators (a reference would be helpful). I'll just point out one possible relation which may just be a ...
Ian Agol's user avatar
  • 66.8k
8 votes
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Axiomatic definition of quantum groups

I would say that if you are looking for a concrete definition then it's better to adopt the Tannakian point of view and to focus on the category of representations of the quantum group rather than on ...
Réamonn Ó Buachalla's user avatar

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