# 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 $\displaystyle B_{n}$ $\displaystyle=\{\pm e_{i}\pm e_{j}\colon 1\leq i $\displaystyle C_{n}$ $\displaystyle=\{\pm e_{i}\pm e_{j}\colon 1\leq i $\displaystyle D_{n}$ $\displaystyle=\{\pm e_{i}\pm e_{j}\colon 1\leq i

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:

 $\displaystyle G_{2}$ $\displaystyle=A_{3}\cup\left\{\pm\frac{1}{3}(2e_{1}-e_{2}-e_{3}),\pm\frac{1}{3% }(-e_{1}+2e_{2}-e_{3}),\pm\frac{1}{3}(-e_{1}-e_{2}+2e_{3})\right\};$ $\displaystyle F_{4}$ $\displaystyle=B_{4}\cup\left\{\frac{1}{2}(\pm e_{1}\pm e_{2}\pm e_{3}\pm e_{4}% )\right\};$ $\displaystyle E_{6}$ $\displaystyle=A_{6}\cup\left\{\pm(e_{7}-e_{8})\}\cup\{\frac{1}{2}(\sum_{i=1}^{% 6}(\pm e_{i})\pm(e_{7}-e_{8}))\colon\text{4 minus signs}\right\};$ $\displaystyle E_{7}$ $\displaystyle=A_{8}\cup\left\{\sum_{i=1}^{8}(\pm e_{i})\colon\text{4 minus % signs}\right\};$ $\displaystyle E_{8}$ $\displaystyle=D_{8}\cup\left\{\sum_{i=1}^{8}(\pm e_{i})\colon\text{even number% of minus signs}\right\}.$ Figure 1: Irreducible root systems and simple Lie algebras.
Title classification of indecomposable root systems ClassificationOfIndecomposableRootSystems 2013-03-22 15:28:56 2013-03-22 15:28:56 rmilson (146) rmilson (146) 6 rmilson (146) Result msc 17B20