WeightLattice 2002-12-05 12:51:17 2007-06-14 10:01:35 Definition integral weight % this is the default PlanetMath 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 The weight lattice $\Lambda_W$ of a root system $R\subset E$ is the lattice $$\Lambda_W=\left\{ e\in E \left| \frac{(e,\alpha)}{(\alpha,\alpha)}\in\mathbb{Z} \text{ for all } r\in R \right. \right\} .$$ Weights which lie in the weight lattice are called {\em \PMlinkescapetext{integral}}. If $R\subset\mathfrak{h}$ is the root system of a semi-simple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$, then $\Lambda_W$ is exactly the set of weights appearing in finite dimensional representations of $\mathfrak{g}$.