classification of indecomposable root systems

There are four infinite families of indecomposable root systems :

	$\displaystyle A_{n}$	$\displaystyle=\{\pm e_{i}\mp e_{j}\colon 1\leq i<j\leq n+1\};$
	$\displaystyle B_{n}$	$\displaystyle=\{\pm e_{i}\pm e_{j}\colon 1\leq i<j\leq n\}\cup\{\pm e_{i}% \colon 1\leq i\leq n\};$
	$\displaystyle C_{n}$	$\displaystyle=\{\pm e_{i}\pm e_{j}\colon 1\leq i<j\leq n\}\cup\{\pm 2e_{i}% \colon 1\leq i\leq n\};$
	$\displaystyle D_{n}$	$\displaystyle=\{\pm e_{i}\pm e_{j}\colon 1\leq i<j\leq n\}$

The subscript on the name of the root system is the dimension of $\mathbf{E}$ , the ambient Euclidean space containing the root system. In the case of $A_{n}$ , the ambient $\mathbf{E}$ is the $n$ -dimensional subspace perpendicular to $\sum_{i=1}^{n}e_{i}$ . In the other 3 cases, $\mathbf{E}=\mathbb{R}^{n}$ . Throughout, we endow $\mathbb{R}^{n}$ with the standard Euclidean inner product, and let $e_{i},\;1\leq i\leq n$ denote the standard basis.

As well, there are 5 exceptional, crystallographic root systems:

	$\displaystyle G_{2}$	$\displaystyle=A_{3}\cup\left\{\pm\frac{1}{3}(2e_{1}-e_{2}-e_{3}),\pm\frac{1}{3% }(-e_{1}+2e_{2}-e_{3}),\pm\frac{1}{3}(-e_{1}-e_{2}+2e_{3})\right\};$
	$\displaystyle F_{4}$	$\displaystyle=B_{4}\cup\left\{\frac{1}{2}(\pm e_{1}\pm e_{2}\pm e_{3}\pm e_{4}% )\right\};$
	$\displaystyle E_{6}$	$\displaystyle=A_{6}\cup\left\{\pm(e_{7}-e_{8})\}\cup\{\frac{1}{2}(\sum_{i=1}^{% 6}(\pm e_{i})\pm(e_{7}-e_{8}))\colon\text{4 minus signs}\right\};$
	$\displaystyle E_{7}$	$\displaystyle=A_{8}\cup\left\{\sum_{i=1}^{8}(\pm e_{i})\colon\text{4 minus % signs}\right\};$
	$\displaystyle E_{8}$	$\displaystyle=D_{8}\cup\left\{\sum_{i=1}^{8}(\pm e_{i})\colon\text{even number% of minus signs}\right\}.$

The following table indicates the cardinality of and the Lie algebras and Dynkin diagrams corresponding to the above root systems.

Figure 1: Irreducible root systems and simple Lie algebras.

Mathworld — Figure 1: Irreducible root systems and simple Lie algebras.

Title	classification of indecomposable root systems
Canonical name	ClassificationOfIndecomposableRootSystems
Date of creation	2013-03-22 15:28:56
Last modified on	2013-03-22 15:28:56
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	6
Author	rmilson (146)
Entry type	Result
Classification	msc 17B20