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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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Quantum groups at roots of 1: problem with ground ring

I must be making a basic error in my reading of Lusztig's Quantum Groups at Roots of 1, and I hope someone can show me what it is. Here is the setup: $v$ is an indeterminate, $\mathbb{Q}(v)$ is the quotient field of $\mathbb{Z}[v, v^{-1}]$, and…
Matthew Towers
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What exactly is a R-matrix?

I am a beginner trying to figure out what is a R-matrix, in relation to the Yang-Baxter Equation. The entry at wikipedia is a little too short. I have read that a solution to the Yang-Baxter equation…
yoyostein
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"The" Universal R Matrix in Quantum Groups

For a quantum group (a quasitriangular Hopf algebra) $A$, it has a distinguished element in $A \otimes A$ called the universal R matrix in many texts (e.g. Kassel). This confuses me, because nowhere do I see any statement about uniqueness of $R$. In…
Aaron
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Does someone recognize this Clebsch-Gordan series?

In short (just the dimensions): $6\cdot6=1+1+6+8+8+12$. Does a Lie group expert recognize that pattern? What's fishy is the second "$1$" (is that allowed by Schur's Lemma if it's an "antisymmetric $1$"?). But otherwise, it looks perfectly like a…
Hauke Reddmann
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questions about quantum groups

I am reading Lusztig's book Introduction to quantum groups. I have a question on page 3. In the fourth line of section 1.2.2, it is said that $'f \otimes 'f$ is associative. I don't know why. I think that $((x_1\otimes x_2)(x'_1\otimes…
LJR
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What are generators and relations for $\hat{\mathfrak{sl}}_2$ and $U_q(\hat{\mathfrak{sl}}_2)$?

We know generators and relations for $\mathfrak{sl}_2$: $e, f, h, [e, f]=h, [h,e]=2e, [h,f]=-2f$. Generators and relations for $U_q(\mathfrak{sl}_2)$ are $e, f, k=q^h$, $kek^{-1}=q^2e, kfk^{-1}=q^{-2}f, [e, f]=\frac{k-k^{-1}}{q-q^{-1}}$. What are…
LJR
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$SU(2) \otimes SU(2)$ and Compact Matrix Quantum Groups

$SU_{\mu}(2)$ (where $\mu \in \mathbb{R}^{+}$) is an example of a compact matrix quantum group, as defined by Woronowicz, but is $SU(2) \otimes SU(2) = Spin (4)$ also a compact matrix quantum group?
Colliding Particles
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A question of the book "a guide to quantum groups"

I am reading this book "a guide to quantum groups" written by V.C. and A.P. But the proof of propersition 4.2.3 on page 121 confused me. Just this place " Applying $id \bigotimes S \bigotimes S^2$ to both sides and reversing the order of the factors…
Xiaosong Peng
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Quantum $SU(2)$ and $3$-sphere

There is a dipheomorphism of $SU(2)$ and $3$-sphere. What happens when you construct quantum $SU(2)$? Do we have an lifting of homeomorphism (or some other morphism) of quantum group to some other object -> quantized 3-sphere?
vejn
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What is an Example of a Co-commutative but not Commutative Quantum Group?

I am looking for 'a' right candidate for an "abelian" quantum group. In a comment to another question it was suggested that the correct candidate was co-commutative. It is straightforward to show that if $G$ is an abelian group then $F(G)$ is…
JP McCarthy
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Why the coproduct of quantum groups are defined in this way?

Let $U_q(g)$ be a quantum group generated by $e_i, f_i, k_{\lambda}$, $\lambda \in Q$, $Q$ is the weight lattice of the Lie algebra $g$. The coproduct of $U_q(g)$ is defined as follows (I only write the formulas for $e_i, f_i$): \begin{align} &…
LJR
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An equation about the Sklyanin bracket

I am reading the lecture http://www.math.uiuc.edu/~ruiloja/Poisson2014/EtingofLectures.pdf. I have a question on page 25. I do not know how to calculate the equation (3.2). Thank you very much for any answers.
Daisy
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Uniqueness of R-matrix

This probably has an easy answer, but is there any sense in which the following statement is true: Let $U$ be a quantum enveloping algebra with universal $R$-matrix denoted by $R$, then $R$ is unique. So I am wondering if (1) there is a suitable…
Elden Elmanto
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Twist map as a solution of the Quantum Yang-Baxter Equation (QYBE)

I am a beginner learning Quantum Groups, I have a question of how to show that that twist map $\tau_{M,M}:M\bigotimes M \rightarrow M\bigotimes M$ is a solution to the QYBE. I tried to prove it by definition: When…
yoyostein
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Definition of quantum group of isometries

Let $(S,\Delta)$ be a compact quantum group and $U\in M(K(H)\otimes S)$ be a unitary corepresentation of $S$ on $H$. Let $\phi $ be a state of $S$. Let $A$ be a sub-$C^*$-algebra of $B(H)$. If $a\in A$, we let $Ad_U(a)=U(a\otimes 1)U^*$. Why the…
Zouba
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