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Book: IntuitingExceptionalRootSystems


See: Math notebook, Classical Lie groups, Classical Lie Root systems, Intuiting classical root systems

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.

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Puslapis paskutinį kartą pakeistas 2018 lapkričio 20 d., 21:53