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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.

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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 $1$"...
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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 ...
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1answer
2k views

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 ...
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1answer
83 views

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 ...
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760 views

Is there some kind of character theory for representations of finite dimensional algebras?

We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with ...
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1answer
561 views

Graded and Filtered Algebras

This is right out of Kassel's Quantum Groups book which I am self-studying. It is on page 14. The general set-up is this. Let $A$ be a filtered algebra with filtration $F_0(A) \subset F_1(A) \subset \...
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What is the antipode in a Combinatorial Hopf Algebra (or graded bialgebra)?

In several papers I've seen on Combinatorial Hopf Algebras, the algebra and coalgebra structures are described, but no antipode is defined. CHAs generally have a natural grading, and are of finite ...
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1answer
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Linear independency of a set of functions.

Let $m\in \mathbb{Z}, \mu_{m}^{j}\in \mathbb{C}, \lambda_{m'}^{j}\in \mathbb{C}, \Psi_{i,r}^{+}\in \mathbb{C}$. $$\lambda^{j}(z)=\sum_{m'\in \mathbb{Z}}\lambda_{m'}^{j}z^{m'}$$ $$z^{-m}...
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1answer
110 views

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 $a\in A$...
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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 ...
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Tensor product decomposion in Lie algebras

As a chemist, I do this all the time...for symmetry groups. Which are finite, luckily :-) For knot theory purposes, I'd like to have a complete list of Lie group irreps R with the property that the ...
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Reference for Quantum groups

I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is ...
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1answer
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Compare of coefficients of two formal power series

Define $$ \Lambda_{i}(u) = \sum_{r=0}^{\infty} \Lambda_{i,r}u^{r}, \Psi_{i}(u) = \sum_{m=0}^{\infty} \psi_{i,m}u^{m}=k_i\frac{\Lambda_{i}(uq_i^{-1})}{\Lambda_{i}(uq_i)}. $$ How can we show that $$ \...
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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 ...
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1answer
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questions about the paper: Affine quivers and canonical bases

I am reading the paper Affine quivers and canonical bases. I have a question on page 114 of the paper. In the proof of property (b), line 6 of page 114, why "for each $\gamma \neq 1$, $tr(\gamma, M)=0$...
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A simple Hopf algebra problems

I have a little question when I read an article. Someone can give me any clue? Let $\mathrm{H}$ be a Hopf algebra and $\mathrm{B}$ be a braided bialgebra in Left-left-Yetter-Drinfled-module over $\...
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1answer
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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. ...
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1answer
162 views

How to compute the Gel'fand Models for a (quantum) Lie Algebra

Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean $\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and $V(\lambda)$...
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1answer
351 views

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, ...
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1answer
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Definition of the Quantum Plane

When ever I find definition of the quantum plane it says $A_q^2 = C\langle x,y \rangle/I$, where $I = C\langle xy-qyx \rangle$. What I want to know is whether they mean the unital free algebra or just ...
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1answer
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Zero divisors in $SU_q(2)$

I'm looking at the quantum group $SU_q(2)$ (over ${\mathbb C}$) and can't see why it has no zero divisors. It's clear that $M_q(2)$, the quantum $2 \times 2$ matrices have no zero divisors, but I can'...
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2answers
615 views

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.