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'weight lattice'
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| Title of object: |
weight lattice |
| Canonical Name: |
WeightLattice |
| Type: |
Definition |
| Created on: |
2002-12-05 12:51:17.585935-05 |
| Modified on: |
2002-12-05 13:02:54.228698-05 |
| Classification: |
msc:17B20 |
Preamble:
% 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 |
Content:
The weight lattice $\Lambda_W$ of a root system $R\subset E$ is the dual lattice to $\Lambda_R$,
the root lattice of $R$. That is, $$\Lambda_W=\{e\in E| (e,r)\in\mathbb{Z}\}$$. Since the simple roots
are free generators of the root lattice, one need only check that $(e,\pi)\in \mathbb{Z}$ for all
simple roots $\pi$. 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 $\mathfrak{g}$. |
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