QuantumGroupsAndHopfAlgebras 2008-09-28 16:11:13 2008-09-28 16:38:27 Topic \med quantum groups and Hopf algebras % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate,color} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} \setlength{\textwidth}{6.5in} %\setlength{\textwidth}{16cm} 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%\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\midsqn}[1]{\ar@{}[dr]|{#1}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \subsection{The $SU(2)$ Quantum Group} Let us consider the structure of the ubiquitous quantum $SU(2)$ group (Woronowicz 1987, Chaician and Demichev 1996). Here $A$ is taken to be a C*--algebra generated by elements $\a$ and $\beta$ subject to the relations: \begin{equation} \begin{aligned} \a \a^* + \mu^2 \beta \beta^* &= 1~,~ \a^* \a + \beta^* \beta = 1~, \\ \beta \beta^* = \beta^* \beta~,~ \a \beta &= \mu \beta \a~,~ \a \beta^* = \mu \beta^* \a~, \\ \a^* \beta = \mu^{-1} \beta \a^*~,~ \a^* \beta^* &= \mu^{-1} \beta^* \a^*~, \end{aligned} \end{equation} where $\mu \in [-1, 1]\backslash \{0\}$~. In terms of the matrix \begin{equation} u = \bmatrix \a & - \mu \beta^* \\ \beta & \a^* \endbmatrix \end{equation} the coproduct $\Delta$ is then given via \begin{equation} \Delta (u_{ij}) = \sum_k u_{ik} \otimes u_{kj}~. \end{equation} \begin{example}\label{hopfalgebra} \emph{The $SL_q (2)$ Hopf algebra.} The Hopf algebra $SL_q (2)$ is defined by the generators $a, b, c, d$ and the following relations: \begin{equation} ba = qab~,~ db=qbd ~,~ ca = qac~,~dc = qcd~,bc = cb~, \end{equation} together with \begin{equation} ad – da = (q^{-1}- q)bc ~,~ad – q^{-1}bc = 1~, \end{equation} and \begin{equation} \Delta \bmatrix a & b \\ c & d \endbmatrix = \bmatrix a & b \\ c & d \endbmatrix \otimes \bmatrix a & b \\ c & d \endbmatrix ~,~ \epsilon \bmatrix a & b \\ c & d \endbmatrix = \bmatrix 1 & 0 \\0 & 1 \endbmatrix ~,~ S \bmatrix a & b \\ c & d \endbmatrix = \bmatrix d & -qb \\-q^{-1}c & a \endbmatrix ~. \end{equation} \end{example}