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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Tensor Product Question in Kassel's Quantum Groups

In Kassel's book on Quantum Groups, I am stuck on the following computation: \begin{eqnarray*} [\Delta (E), \Delta (F)] &=& \Delta (E)\Delta (F)-\Delta (F)\Delta (E)\\ &=& (1\otimes E ...
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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}...
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Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$.

I read a textbook on Quantum Groups by Kassel, and did not understand the following proof of well-definedness of $\Delta$ (comult.) and $\epsilon$ (counit) on $GL_q(2)$ and $SL_q(2)$: (pg 84) The ...
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how to calculate the derivative of a plane wave in non commutative geometry

Shahn Majid and Eliezer Batista find the derivative of a wave plane in their paper: Non Commutative Geometry of Angular Momentum Space $U \mathfrak{su}(2)$. They obtained the non commutating ...
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Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra

As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
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The longest word in Weyl group and positive roots.

How to write down a reduced decomposition of the longest word in a Weyl group? For example, how to write down a reduced decomposition of the longest word in type B3 Weyl group? For a decomposition of ...
3
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1answer
100 views

proving that this coproduct and product get along well enough to make a bialgebra (Quantum Groups, Kassel)

I'm working through Kassel's Quantum Groups book and I'm stuck on what looks like a pretty simple problem (ch3, ex 2 in the book): Let $k$ be a field and $C= k[t]$ be a coalgebra with coproduct: $$ \...
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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 ...
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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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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 $1$"...
4
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1answer
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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 ...
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2answers
769 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 ...
3
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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 ...
4
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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 ...
5
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1answer
164 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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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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1answer
216 views

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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616 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.
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160 views

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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1answer
186 views

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
249 views

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'...