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Questions tagged [quantum-groups]

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure.

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. Reference: Wikipedia.

In general, a quantum group is some kind of Hopf algebra. There is no single, all-encompassing definition, but instead a family of broadly similar objects.

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What are Quantum Groups?

I am going to do a semester project (kind of a little thesis) this spring. I met a professor and asked him about some possible arguments. Among others, he proposed something related to quantum groups. I am utterly ignorant of the subject, so could…
Daniel Robert-Nicoud
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Why study Hopf Algebras?

I'm looking for reasons that motivate the study of Hopf Algebra, like its applications in other branches of mathematics or maybe with physics. The first I've got is that they're interesting by themselves. But I'll want to motivate the undergaduate…
Math.mx
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Why is the Quasitriangular Hopf algebra called "Quasitriangular"?

The precise definition of a Quasitriangular Hopf algebra can be found on wikipedia. What is the reason behind the word "Quasitriangular"? Is it because the R-matrix is a triangular matrix, or is it related to some properties of triangles? Thanks…
yoyostein
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What exactly is a tensor product?

This is a beginner's question on what exactly is a tensor product, in laymen's term, for a beginner who has just learned basic group theory and basic ring theory. I do understand from wikipedia that in some cases, the tensor product is an outer…
yoyostein
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Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma you obtain QGamma[n+1,q] / QGamma[n,q] = (q^n-1) /…
André
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Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, if we let $S_n$ act on $(\mathbb{C}^N)^{\otimes…
Daniel McLaury
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Duality between universal enveloping algebra and algebras of functions

There is a well known duality (of Hopf algebras) between universal enveloping algebra $U(\mathfrak{g})$ of a complex Lie algebra $\mathfrak{g}$ of a compact group $G$ and the algebra of continuous functions $C(G)$. My question is, is there in the…
Henrique Tyrrell
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Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will this decomposition obey Krull-Schmidt, by which…
Dyke Acland
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Why are Hopf algebras called quantum groups?

Why are noncommutative nonassociative Hopf algebras called quantum groups? This seems to be a purely mathematical notion and there is no quantum anywhere in it prima facie.
user218
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A little bit of Intuition for Corepresentations from Representations

Hi folks I am trying to prove what I think should be a straightforward enough result but I am having to make a somewhat unnatural definition to do it. This unnatural definition is hinted at in a paper by Franz & Gohm: If $G$ is a group, then…
JP McCarthy
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Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a classical system, one…
N.U.
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The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional subspace $$W=\{\lambda(1,1,1,1,I_{2\times…
JP McCarthy
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Dual of a Hopf algebra

Given is a Hopf algebra $(H,m,\eta, \Delta, \epsilon, S)$. We know that there is a dual notion of it, called the dual Hopf algebra on $H^{*}$ as a vector space. It has the natural structure of a Hopf algebra. We know that the finite-dimensional…
Lullaby
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Some concrete examples of $M_q(2)$ points

Given $q \in \mathbb{C}$ invertible, Kassel says that an $M_q(2)$ point of an $R$ algebra is a $m=\left(\begin{array}{cc} A & B\\ C & D \end{array}\right)\in R^{4}$ such that $A,\,B,\,C,\,D \in R$ satisfy the following relations $$CA=qAC,$$$$…
Dac0
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Basis of $SL_q(2)$

While trying to show that $SL_q(2)$ is noncocommutative, I needed to prove the following fact: Show that the set $\{a^ib^jc^k\}_{i,j,k\geq 0}\cup\{b^ic^jd^k\}_{i,j\geq 0,k>0}$ is a basis of $SL_q(2)$. I was stuck with proving the above fact.…
yoyostein
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