Articles of quantum groups

$U_q$ 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{eqnarray*} &&KK^{-1}=K^{-1}K=1\\ &&KEK^{-1}=q^2 E,\ KFK^{-1}=q^{-2} F\\ and &&[E,F]=\frac{K-K^{-1}}{q-q^{-1}} \end{eqnarray*}” May I ask if there is a way of understanding what are $E, F, K$ and $K^{-1}$? […]

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 the longest word, how can we write down an ordering of positive roots? I am asking these questions […]

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 equation is motivated by the fact that the differentiation operator $T=d/dx$ on complex coalgebra $\mathbb{C}[x]$ satisfies this equation.

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 $$W=\{\lambda(1,1,1,1,I_{2\times 2}):\lambda\in\mathbb{C}\}.$$ By my reckoning there should be seven more one dimensional invariant subspaces and hence irreducible corepresentations. The eight-dimensional Kac-Paljutkin Quantum Group Here we give the defining […]

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 is the general formula for multiplying and dividing quantum integers? This is probably well-known but I don’t have a reference. For example, we have $$ [2]_q[2]_q = [3]_q+1, […]

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 product, which takes two vectors, say $\textbf{u}$ and $\textbf{v}$, and outputs […]

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 someone help me out and tell me What is a quantum […]

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 I mean will the type and multiplicity of the irreducible comodules appearing be the same in any […]

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: $$A(0,v)=\left(\begin{array}{cc}1+\zeta^{-v} & \zeta^v+\zeta^{2v}\\ \zeta^{-v}+\zeta^{-2v}&1+\zeta^{v}\end{array}\right),$$ $$A(1,v)=\left(\begin{array}{cc}\zeta^{-1}+\zeta^{-v} & \zeta^{v}\\ \zeta^{-v}&\zeta^{-1}+\zeta^{v}\end{array}\right).$$ $$A(n-1,v)=\left(\begin{array}{cc}\zeta+\zeta^{-v} & \zeta^{2v}\\ \zeta^{-2v}&\zeta+\zeta^v\end{array}\right).$$ I am hoping to calculate for each of these $A$ $$\text{Tr}\left[\left(A^k\right)^*A^k\right]=\text{Tr}\left[\left(A^*\right)^kA^k\right].$$ All I have is that $A$ and $A^*$ in general do not commute so I can’t simultaneously diagonalise […]