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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Intuit the five exceptional root systems.

The four classical root systems are as follows, where throughout, {$i>j$}:

 {$A_n$} {$\pm (x_i-x_j)$} {$B_n$} {$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i$} {$C_n$} {$\pm (x_i-x_j), \pm (x_i+x_j), \pm 2x_i$} {$D_n$} {$\pm (x_i-x_j), \pm (x_i+x_j)$}

The five exceptional root systems are described in Wikipedia's article on root systems.

 {$G_2$} {$(x_i-x_j), (x_i - x_j) - (x_j - x_k)$} for all distinct {$i,j,k \in \{1,2,3\}$}.

Thus {$G_2$} is the disjoint union of two copies of {$A_2$}.

Alternatively,

 {$G_2$} {$\pm (x_i-x_j)$} where {$i \neq j$}, and {$\pm (3x_i - (x_1+x_2+x_3))$} for all {$i$}.
 {$F_4$} {$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i, \frac{1}{2}(\pm x_1 \pm x_2 \pm x_3 \pm x_4)$}

Thus {$F_4$} has a copy of {$B_4$} as a subset.

 {$E_8$} {$\pm (x_i-x_j), \pm (x_i+x_j)$}, and {$\frac{1}{2}(\sum_{i=1}^{8}(-1)^{a_i}x_i)$} where {$\sum_{i=1}^{8}{a_i} \in 2\mathbb{Z}$}.

{$E_7$} and {$E_6$} are subsets of {$E_8$}.

Literature

Notes by Lecture 16 on Lie Algebras by Kevin McGerty.

#### IntuitingExceptionalRootSystems

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