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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4k views

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. ...
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4answers
2k views

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 ...
13
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2answers
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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 ...
12
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0answers
311 views

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 ...
12
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1answer
354 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, ...
9
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1answer
126 views

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 ...
8
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2answers
619 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.
8
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1answer
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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 ...
7
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3answers
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Concrete cases where $YX=qXY$

I was reading Kassel on the quantum plane and he defines an $R$-point on this plane as a pair of $X$, and $Y$ elements of the non commutative algebra $R$ such that $$YX=qXY,$$ with $q$ invertible. Can ...
7
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548 views

arithmetic with quantum integers

Consider the ring $\mathbb{Z}[q^{\pm 1}]$. For $n \in \mathbb{N}$, define the quantum integers: $$[n]_q := \frac{q^n-q^{-n}}{q-q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-(n-3)} + q^{-(n-1)}$$ What ...
7
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1answer
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Quantum Groups: prove $U_1'\cong U[K]/(K^2-1)$ and $U\cong U_1'/(K-1)$

Regarding this theorem, which is in Kassel pg 126, I have two questions. I have typed in the relevant material for reference. 1) How does $$U_1'\cong U[K]/(K^2-1)$$ imply $$U\cong U_1'/(K-1)$$? ...
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A question on coalgebras(1)

Is there a complex coalgebra $C$ with dimension at least 2 for which the scalar operators $T(x)=\lambda x$ are the only operators which satisfy $$(T\otimes T)\circ \Delta= \Delta \circ T^{2}$$ This ...
7
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1answer
228 views

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 ...
6
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4answers
994 views

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 ...
6
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1answer
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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 ...
6
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1answer
238 views

The “$\mathbb{Z}_n$-theta function” - what is it? Is it being studied somewhere?

The Jacobi theta function is well known: $$\theta(z, \tau) = \sum_{n=-\infty}^\infty \mathrm{e}^{\pi i n^2 \tau + 2\pi i nz}$$ In Shahn Majid's "Foundations of Quantum Group Theory", you'll find a ...
6
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1answer
194 views

A question on coalgebras(2)

Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order comultiplication can be defined inductively as follows(with some abuse of notations we denote them by ...
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Some concrete examples of $M_q(2)$ points

Given $q \in C$ invertible, Kassel says that a $M_q(2)$ point of an $R$ algebra is a $m=\left(\begin{array}{cc} A & B\\ C & D \end{array}\right)\in R^{4}$ such that $A,\,B,\,C,\,D \in R$ ...
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Reconstruction of monoidal categories

Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator). Bruce Westbury ...
6
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1answer
270 views

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 ...
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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 $...
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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 $V(\lambda)$...
5
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1answer
301 views

From Hopf Algebras to quantum groups

I start with self study about quantum groups. Until now I covered Hopf $*$-algebras and there representations (for example by the book of Klymik and Schmüdgen). Now I want to understand the step from ...
5
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1answer
651 views

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 ...
5
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2answers
255 views

Trace of Product of Powers of $A$ and $A^\ast$

Let $n$ be odd, $\displaystyle v=1,...,\frac{n-1}{2}$ and $\displaystyle \zeta=e^{2\pi i/n}$. Define the following matrices: $$A(0,v)=\left(\begin{array}{cc}1+\zeta^{-v} & \zeta^v+\zeta^{2v}\\ \...
5
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1answer
211 views

Towards a Quantum Peter Weyl Theorem

This is taken from Timmermann's Invitation to Quantum Groups and Duality. Let $(A,\Delta)$ be a *-Hopf algebra and let $\chi:V\rightarrow V\otimes A$ be a corepresentation of $(A,\Delta)$ on a vector ...
5
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141 views

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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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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162 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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Converse of Hopf Algebra Theorem

There is a theorem that if a Hopf algebra $H$ is commutative or cocommutative, then $S^2=id_H$, where $S$ denotes the antipode. May I know if the converse is true? (i.e. if $S^2=id_H$, does it ...
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 ...
4
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2answers
213 views

If $H$ is a Hopf algebra, do we have $H^{cop}$ is a Hopf algebra?

Let $H=(H, m, u, \Delta, \epsilon, S)$ be a Hopf algebra, see for example the lecture notes, where $m$ is the multiplication, $u$ is the unit, $\Delta$ is the comultiplication, $\epsilon$ is the ...
4
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2answers
188 views

A question on a Hopf algebra

Let $(H, \nu,\eta, \Delta, \epsilon, S)$ be a Hopf algebra. $S$ is the antipode. I am reading a proof of the fact $S(xy)=S(y)S(x)$. First, define maps $\nu, \rho$ in $\hom(H \otimes H, H)$ by $\nu(...
4
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2answers
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$U_q$ Quantum group and the four variables: $E, F, K, K^{-1}$

In Kassel's book on Quantum groups, it is defined that: "We define $U_q=U_q(\mathfrak{sl}(2) )$ as the algebra generated by the four variables $E$, $F$, $K$, $K^{-1}$ with the relations \begin{...
4
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1answer
82 views

Why do we need the condition “cocommutative” in the definition of a coPoisson Hopf algebra?

In this paper, page 5, Section 3.6, in the definition of a coPoisson Hopf algebra $H$, it is said that: a coPoisson Hopf algebra is a cocommutative Hopf algebra $A$ with a map $\delta: A \to A \otimes ...
4
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1answer
85 views

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 ...
4
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1answer
1k views

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 ...
4
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1answer
426 views

Definition of the Quantum plane and the Yang Baxter Equation

I was reading this on the quantum plane and the Yang Baxter equation. John Baez says that imposing $$ R(X\otimes X)= X\otimes X $$ $$ R(Y\otimes Y)= Y\otimes Y $$ $$ R(X\otimes Y)=q Y\otimes X $$ $$ R(...
4
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1answer
137 views

Graphical calculus for braided Hopf algebras

I am trying to understand the graphical calculus presented in Ohtsuki's book Quantum Invariants. I think if I understand these first few examples it will help me greatly. Let $(A,m,i,\Delta,\epsilon,...
4
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1answer
115 views

Center of a quantum matrix algebra

Let $p \in k^\times$ be a nonroot of unity. It seems to be a well-known fact that the center of the quantum matrix algebra $\mathcal{O}_p(M_n(k))$ is generated by the quantum determinant $D_p$. It is ...
4
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1answer
54 views

Endomorphism ring of an irreducible comodule

Given a Hopf algebra over $\mathbb{C}$, and a irreducible comodule $V$, I can show that Mor$(V,V) \simeq \mathbb{C}$, where Mor$(V,V)$ is the space of comodule maps from $V$ to $V$. My proof uses that ...
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Toy example of deformed diffeomorphism group

Consider a toy example of a diffeomorphism group – the group of diffeomorphisms of a 1-dimensional manifold with a disconnected boundary (2 points). The group is a group of monotonically increasing ...
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Where does the condition $\pi(X_-)+\pi(X_+)=0$ in dual Lie algebra come from?

I am trying to understand the dual Lie algebra in a Lie bialgebra. In the above article, it is said that: "Let ${\displaystyle {\mathfrak {g}}}$ be any semisimple Lie algebra. To specify a Lie ...
4
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109 views

What is the relation between the construction of Reshetikhin-Turaev and Witten's paper?

I have read that the papers of Reshetikhin and Turaev provide a mathematically rigorous framework for the ideas contained in Witten's paper "Quantum field theory and the Jones polynomial". Can ...
4
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0answers
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~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
4
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0answers
161 views

Where do I go from Linear algebra past Calc III to try to learn complex physics (relativity and quantum group theory)?

I'm mainly a programmer, but I have a love for Mathematics that's been, well, insatiable. I've had my eye on learning Quantum Groups and Relativity, but I want to stay in something I can do with ...
4
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0answers
73 views

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 $1$"...
3
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2answers
777 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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1answer
150 views

Primitive elements of finite dimensional Hopf algebras

An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that: "If $H$ is a $\...
3
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1answer
99 views

Alternative Cotensor Definition

Let $H$ be a Hopf algebra, and $(M,\rho_r)$ and $(N,\rho_l)$ right and left $H$-comodules respectively. As usual, we define their cotensor product to be $$ M \square_H N := \text{ker}\{(\rho_r \...