ClassificationOfFiniteDimensionalRepresentationsOfSemiSimpleLieAlgebras 2002-12-04 16:19:21 2007-03-02 13:15:53 Definition highest weight highest vector vector of highest weight highest weight representation % 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 If $\mathfrak{g}$ is a semi-simple Lie algebra, then we say that an representation $V$ has highest weight $\lambda$, if there is a vector $v\in V_\lambda$, the weight space of $\lambda$, such that $Xv=0$ for $X$ in any positive root space, and $v$ is called a {\em highest vector}, or {\em vector of highest weight}. There is a unique (up to isomorphism) irreducible finite dimensional representation of $\mathfrak{g}$ with highest weight $\lambda$ for any dominant weight $\lambda\in\Lambda_W$, where $\Lambda_W$ is the weight lattice of $\mathfrak{g}$, and every irreducible representation of $\mathfrak{g}$ is of this type.