55
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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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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 $q$-...
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18
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 ...
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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\...
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13
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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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13
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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13
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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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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 ...
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11
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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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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 ...
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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 ...
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11
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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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11
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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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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 $...
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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 ...
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10
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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).
...
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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 ...
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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 ...
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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 ...
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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 ...
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9
votes
Quantum double vs Quantum group
A "quantum group" is a somewhat vague term. I can talk a bit about one class of examples, but there are others. Some standard references are Quantum Groups by Kassel and A Guide to Quantum ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
8
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Finite compact quantum groups
Another example apart from the example of Kac & Paljutkin (reference below) are the quantum groups of Sekine:
Y. Sekine, An example of finite-dimensional Kac algebras of
Kac-Paljutkin type, Proc. ...
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8
votes
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What is the discrete quantum group associated to a compact group?
I believe that really the question is being asked in the context of Locally compact quantum groups. This is a framework using the machinery of $C^*$ and von Neumann algebras, with (amoung many aims, ...
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