TopologicalTransformationGroup 2007-02-23 10:53:31 2007-02-23 20:10:43 Definition effective topological transformation group \usepackage{amssymb,amscd} \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} \usepackage{pst-plot} \usepackage{psfrag} % define commands here \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} Let $G$ be a topological group and $X$ any topological space. We say that $G$ is a \emph{topological transformation group} of $X$ if $G$ \emph{acts} on $X$ continuously, in the following sense: \begin{enumerate} \item there is a continuous function $\alpha:G\times X\to X$, where $G\times X$ is given the product topology \item $\alpha(1,x)=x$, and \item $\alpha(g_1g_2,x)=\alpha(g_1,\alpha(g_2,x))$. \end{enumerate} The function $\alpha$ is called the \emph{(left) action} of $G$ on $X$. When there is no confusion, $\alpha(g,x)$ is simply written $gx$, so that the two conditions above read $1x=x$ and $(g_1g_2)x=g_1(g_2x)$. If a topological transformation group $G$ on $X$ is \emph{effective}, then $G$ can be viewed as a group of homeomorphisms on $X$: simply define $h_g:X\to X$ by $h_g(x)=gx$ for each $g\in G$ so that $h_g$ is the identity function precisely when $g=1$. \textbf{Some Examples.} \begin{enumerate} \item Let $X=\mathbb{R}^n$, and $G$ be the group of $n\times n$ matrices over $\mathbb{R}$. Clearly $X$ and $G$ are both topological spaces with the usual topology. Furthermore, $G$ is a topological group. $G$ acts on $X$ continuous if we view elements of $X$ as column vectors and take the action to be the matrix multiplication on the left. \item If $G$ is a topological group, $G$ can be considered a topological transformation group on itself. There are many continuous actions that can be defined on $G$. For example, $\alpha:G\times G\to G$ given by $\alpha(g,x)=gx$ is one such action. It is continuous, and satisfies the two action axioms. $G$ is also effective with respect to $\alpha$. \end{enumerate}