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Book: LieAlgebrasChoiceTemplates


See: Lie theory, Classical Lie groups

Interpret the four classical Lie algebras in terms of the four choice templates.


There are four choice templates:

The root systems of the classical Lie algebras all include the roots of {$A_{n}$}: {$\pm (x_i-x_j)$}. The other root systems all include the additional roots of {$D_{n}$}: {$\pm (x_i+x_j), i\neq j$}. Finally, {$B_{n}$} includes {$\pm x_i$} and {$C_{n}$} includes {$\pm (x_i+x_i)$}.

The Weyl group of a root system is the group generated by the reflections across the hyperplanes of the roots. So the Weyl group is given by the directions of the roots but not their lengths.

Hyperplane = {$\pm$} RootTransposition
{$\pm(e_2-e_1)$}{$e_1 \Leftrightarrow e_2, -e_1 \Leftrightarrow -e_2$}
{$\pm(e_2+e_1)$}{$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}
{$\pm(e_1)$}{$e_1 \Leftrightarrow -e_1$}

There are three ways of defining the reflections {$e_1 \Leftrightarrow -e_1$}:

Pairing template

The sign can be attributed to the element or to the pairing template. So try to think that through.

Consider {$X-Y$} as a template where the minus sign imposes an ordering.

Consider {${X,Y}$} as a template where the set indicates inclusion with no ordering.

Not satisfactory. :(

Orthogonalities

Readings



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Puslapis paskutinį kartą pakeistas 2019 balandžio 30 d., 14:23