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[parent] classification of indecomposable root systems (Result)

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:

$\displaystyle G_2$ $\displaystyle =A_3 \cup \left\{ \pm\frac{1}{3}( 2 e_1 -e_2-e_3), \pm\frac{1}{3}( - e_1 +2e_2-e_3),\pm\frac{1}{3}( - e_1 -e_2+2e_3)\right\};$    
$\displaystyle F_4$ $\displaystyle =B_4 \cup \left\{ \frac{1}{2}(\pm e_1\pm e_2\pm e_3 \pm e_4)\right\};$    
$\displaystyle E_6$ $\displaystyle = A_6 \cup \left\{ \pm(e_7-e_8) \} \cup \{ \frac{1}{2}( \sum_{i=1}^6 (\pm e_i) \pm (e_7 - e_8)) \colon \text{4 minus signs}\right\};$    
$\displaystyle E_7$ $\displaystyle =A_8\cup \left\{\sum_{i=1}^8 (\pm e_i) \colon \text{4 minus signs}\right\};$    
$\displaystyle E_8$ $\displaystyle =D_8\cup \left\{\sum_{i=1}^8 (\pm e_i) \colon \text{even number of minus signs}\right\}.$    

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

Figure: Irreducible root systems and simple Lie algebras.
\includegraphics{rootsys-diag}



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Cross-references: Dynkin diagrams, Lie algebras, cardinality, crystallographic, standard basis, inner product, Euclidean, perpendicular, subspace, Euclidean space, dimension, subscript, root systems, indecomposable, infinite

This is version 3 of classification of indecomposable root systems, born on 2005-08-20, modified 2006-04-10.
Object id is 7337, canonical name is ClassificationOfIndecomposableRootSystems.
Accessed 1557 times total.

Classification:
AMS MSC: 17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive )
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