find the non-compact simple root associated to a given non-compact root in the Satake diagram - Maple Programming Help

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LieAlgebras[SatakeAssociate] - find the non-compact simple root associated to a given non-compact root in the Satake diagram

Calling Sequences

SatakeAssociate( )

Parameters

$\mathrm{α}$     - a column vector, a non-compact root of a non-compact, simple Lie algebra

$\mathrm{Δ0}$    - a list of column vectors, the simple roots of a non-compact simple Lie algebra

$\mathrm{Δc}$    - (optional) a list of column vectors, defining the compact roots of non-compact simple Lie algebra

Description

 • Let Δ be the root system for a non-compact, simple Lie algebra. Let ${\mathrm{Δ}}^{+}$be a set of positive roots, the compact roots, ${\mathrm{Δ}}_{0}$be the simple roots and the compact simple roots. We chose the positive roots to be closed under complex conjugation. Then for each root there is a unique root such that The root is called the Satake associate of ${\mathrm{α}}_{}$.
 • The command SatakeAssociate( ) returns the Satake associate of $\mathrm{α}$.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

Here is the Satake diagram for and the corresponding simple roots.

 > SatakeDiagram("su(4, 4");
 > alpha1, alpha2, alpha3, alpha4, alpha5, alpha6, alpha7 := seq(Vector(v), v = [[-2*I, I, 0, 1, -1, 0, 0], [I, -2*I, I, 0, 1, -1, 0], [0, I, -2*I, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 2], [0, -I, 2*I, 0, 0, 1, -1], [-I, 2*I, -I, 0, 1, -1, 0], [2*I, -I, 0, 1, -1, 0, 0]]);
 ${\mathrm{α1}}{,}{\mathrm{α2}}{,}{\mathrm{α3}}{,}{\mathrm{α4}}{,}{\mathrm{α5}}{,}{\mathrm{α6}}{,}{\mathrm{α7}}{:=}\left[\begin{array}{c}{-}{2}{}{I}\\ {I}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {-}{2}{}{I}\\ {I}\\ {0}\\ {1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {I}\\ {-}{2}{}{I}\\ {0}\\ {0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {-}{I}\\ {2}{}{I}\\ {0}\\ {0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {2}{}{I}\\ {-}{I}\\ {0}\\ {1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {-}{I}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]$ (2.1)
 > Delta0 := [alpha1, alpha2, alpha3, alpha4, alpha5, alpha6, alpha7]:

All the roots are non-compact so that the Satake associate is just the complex conjugate, for example,

 su44 > alpha1, SatakeAssociate(alpha1, Delta0), alpha7;
 $\left[\begin{array}{c}{-}{2}{}{I}\\ {I}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {-}{I}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {-}{I}\\ {0}\\ {1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]$ (2.2)

The root is its own associate.

 > SatakeAssociate(alpha4, Delta0), alpha4;
 $\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\\ {2}\end{array}\right]$ (2.3)

Example 2

Here is the Satake diagram for and the corresponding simple roots.

 > SatakeDiagram("so(7,2)");
 > alpha1, alpha2, alpha3, alpha4 := seq(Vector(v), v = [[1, -1, 0, 0], [0, 1, -I, 0], [0, 0, I, -I], [0, 0, 0, I]]);
 ${\mathrm{α1}}{,}{\mathrm{α2}}{,}{\mathrm{α3}}{,}{\mathrm{α4}}{:=}\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {I}\end{array}\right]$ (2.4)
 > Delta0 := [alpha1, alpha2, alpha3, alpha4]:

Roots and are compact. The root is real and is therefore its own Satake associate. The root satisfies

 > map(conjugate, alpha2) - alpha2, 2*alpha3 + 2*alpha4;
 $\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]$ (2.5)

and is therefore also its own Satake associate.

Example 3.

Here is the Satake diagram for and the corresponding simple roots.

 > SatakeDiagram("so(5, 3)");
 > alpha1, alpha2, alpha3, alpha4 := seq(Vector(v), v = [[1, -1, 0, 0], [0, 1, -1, 0], [0, 0, 1, -I], [0, 0, 1, I]]);
 ${\mathrm{α1}}{,}{\mathrm{α2}}{,}{\mathrm{α3}}{,}{\mathrm{α4}}{:=}\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {I}\end{array}\right]$ (2.6)
 > Delta0 := [alpha1, alpha2, alpha3, alpha4]:

There are no compact roots. The roots and are real and therefore are their own Satake associates. Because there are no compact roots the Satake associate of is its complex conjugate which is ${\mathrm{α}}_{4.}$

 > alpha3, SatakeAssociate(alpha3, Delta0), alpha4;
 $\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\\ {I}\end{array}\right]$ (2.7)

Example 4.

Here is the Satake diagram for and the corresponding simple roots.

 > SatakeDiagram("so*(10)");
 > alpha1, alpha2, alpha3, alpha4, alpha5 := seq(Vector(v), v = [[0, 0, 2*I, 0, 0], [1, -1, -I, -I, 0], [0, 0, 0, 2*I, 0], [0, 1, 0, -I, -I], [0, 1, 0, -I, I]]);
 ${\mathrm{α1}}{,}{\mathrm{α2}}{,}{\mathrm{α3}}{,}{\mathrm{α4}}{,}{\mathrm{α5}}{:=}\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {-}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {-}{I}\\ {I}\end{array}\right]$ (2.8)

The roots and are compact. Since

 > map(conjugate, alpha2) - alpha2 = alpha1 + alpha3;
 $\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {2}{}{I}\\ {0}\end{array}\right]{=}\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {2}{}{I}\\ {0}\end{array}\right]$ (2.9)

the Satake associate of  is itself. Since

 > map(conjugate, alpha4) - alpha5 = alpha3;
 $\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{=}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]$ (2.10)

the Satake associate of is

These calculations agree with the output of the command SatakeAssociate.

 > alpha2, SatakeAssociate(alpha2, [alpha1, alpha2, alpha3, alpha4, alpha5]);
 $\left[\begin{array}{c}{1}\\ {-}{1}\\ {-}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {-}{I}\\ {-}{I}\\ {0}\end{array}\right]$ (2.11)
 > alpha4, SatakeAssociate(alpha4, [alpha1, alpha2, alpha3, alpha4, alpha5]), alpha5;
 $\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {-}{I}\\ {I}\end{array}\right]$ (2.12)