find the simple roots for a set of positive roots - Maple Programming Help

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LieAlgebras[SimpleRoots] - find the simple roots for a set of positive roots

Calling Sequences

SimpleRoots(PR$)$

Parameters

PR    - a list of  vectors, giving the positive roots of a simple Lie algebra

Description

 • Let be a list of roots for either an abstract root system or for a simple Lie algebra. In particular, must have an even number of elements and if then . Write  where, if then and then The set ${\mathrm{Δ}}^{+}$is called the set of positive roots .The choice of positive roots is not unique. If ${\mathrm{\Delta }}^{+}$ is set of positive roots,then a root is called a simple root if it is not a sum of any other 2 positive roots. If is a set of simple roots for ${\mathrm{Δ}}^{+}$, then every root in is a linear combination of the roots in ${\mathrm{Δ}}_{0}$ with positive integer coefficients.The number of simple roots equals the rank of the Lie algebra.
 • The command SimpleRoots(PR) returns a list of vectors defining a set of simple roots.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We calculate the simple roots for the Lie algebra This is the 36-dimensional Lie algebra of $8×8$ matrices $A$ which are skew-symmetric with respect to the skew form

We use the command SimpleLieAlgebraData to obtain the structure equations for this Lie algebra.

 > LD := SimpleLieAlgebraData("sp(8, R))", sp8R, labelformat = "gl", labels = ['E', 'omega']):
 > DGsetup(LD);
 ${\mathrm{Lie algebra: sp8R}}$ (2.1)

The following diagonal elements define a Cartan subalgebra. (This can be calculated using the command CartanSubalgebra).

 sp8R > CSA_sp8R := [E11, E22, E33, E44];
 ${\mathrm{CSA_sp8R}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E44}}\right]$ (2.2)

Here is the corresponding root space decomposition.

 sp8R > RSD_sp8R := RootSpaceDecomposition(CSA_sp8R);
 ${\mathrm{RSD_sp8R}}{:=}{\mathrm{table}}\left(\left[\left[{1}{,}{0}{,}{1}{,}{0}\right]{=}{\mathrm{E17}}{,}\left[{0}{,}{-}{1}{,}{0}{,}{1}\right]{=}{\mathrm{E42}}{,}\left[{0}{,}{0}{,}{1}{,}{1}\right]{=}{\mathrm{E38}}{,}\left[{1}{,}{0}{,}{0}{,}{-}{1}\right]{=}{\mathrm{E14}}{,}\left[{-}{2}{,}{0}{,}{0}{,}{0}\right]{=}{\mathrm{E51}}{,}\left[{0}{,}{0}{,}{0}{,}{2}\right]{=}{\mathrm{E48}}{,}\left[{0}{,}{0}{,}{0}{,}{-}{2}\right]{=}{\mathrm{E84}}{,}\left[{0}{,}{0}{,}{2}{,}{0}\right]{=}{\mathrm{E37}}{,}\left[{0}{,}{1}{,}{1}{,}{0}\right]{=}{\mathrm{E27}}{,}\left[{0}{,}{0}{,}{1}{,}{-}{1}\right]{=}{\mathrm{E34}}{,}\left[{-}{1}{,}{0}{,}{1}{,}{0}\right]{=}{\mathrm{E31}}{,}\left[{0}{,}{0}{,}{-}{2}{,}{0}\right]{=}{\mathrm{E73}}{,}\left[{0}{,}{-}{2}{,}{0}{,}{0}\right]{=}{\mathrm{E62}}{,}\left[{-}{1}{,}{0}{,}{0}{,}{-}{1}\right]{=}{\mathrm{E54}}{,}\left[{0}{,}{-}{1}{,}{1}{,}{0}\right]{=}{\mathrm{E32}}{,}\left[{0}{,}{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{E24}}{,}\left[{0}{,}{0}{,}{-}{1}{,}{-}{1}\right]{=}{\mathrm{E74}}{,}\left[{1}{,}{0}{,}{0}{,}{1}\right]{=}{\mathrm{E18}}{,}\left[{0}{,}{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{E23}}{,}\left[{2}{,}{0}{,}{0}{,}{0}\right]{=}{\mathrm{E15}}{,}\left[{1}{,}{0}{,}{-}{1}{,}{0}\right]{=}{\mathrm{E13}}{,}\left[{0}{,}{-}{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{E64}}{,}\left[{0}{,}{2}{,}{0}{,}{0}\right]{=}{\mathrm{E26}}{,}\left[{1}{,}{-}{1}{,}{0}{,}{0}\right]{=}{\mathrm{E12}}{,}\left[{1}{,}{1}{,}{0}{,}{0}\right]{=}{\mathrm{E16}}{,}\left[{0}{,}{1}{,}{0}{,}{1}\right]{=}{\mathrm{E28}}{,}\left[{-}{1}{,}{-}{1}{,}{0}{,}{0}\right]{=}{\mathrm{E52}}{,}\left[{0}{,}{-}{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{E63}}{,}\left[{-}{1}{,}{1}{,}{0}{,}{0}\right]{=}{\mathrm{E21}}{,}\left[{-}{1}{,}{0}{,}{-}{1}{,}{0}\right]{=}{\mathrm{E53}}{,}\left[{-}{1}{,}{0}{,}{0}{,}{1}\right]{=}{\mathrm{E41}}{,}\left[{0}{,}{0}{,}{-}{1}{,}{1}\right]{=}{\mathrm{E43}}\right]\right)$ (2.3)

We calculate the positive roots for .

 sp8R > PR_sp8R := PositiveRoots(RSD_sp8R, <1, 2, 3, 4>);
 ${\mathrm{PR_sp8R}}{:=}\left[\left[\begin{array}{r}{1}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {2}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {2}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {-}{1}\\ {1}\end{array}\right]\right]$ (2.4)

The rank of is 4 so we should find 4 positive roots.

 sp8R > SR_sp8R := SimpleRoots(PR_sp8R);
 ${\mathrm{SR_sp8R}}{:=}\left[\left[\begin{array}{r}{0}\\ {-}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {-}{1}\\ {1}\end{array}\right]\right]$ (2.5)

We check that the positive roots are positive integer linear combinations of the simple roots with the GetComponents command.

 sp8R > GetComponents(PR_sp8R, SR_sp8R);
 $\left[\left[{1}{,}{1}{,}{1}{,}{0}\right]{,}\left[{1}{,}{0}{,}{0}{,}{1}\right]{,}\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}\left[{2}{,}{1}{,}{2}{,}{2}\right]{,}\left[{2}{,}{1}{,}{2}{,}{0}\right]{,}\left[{1}{,}{1}{,}{2}{,}{0}\right]{,}\left[{1}{,}{0}{,}{1}{,}{0}\right]{,}\left[{1}{,}{0}{,}{0}{,}{0}\right]{,}\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}\left[{0}{,}{1}{,}{0}{,}{0}\right]{,}\left[{0}{,}{1}{,}{2}{,}{0}\right]{,}\left[{0}{,}{1}{,}{1}{,}{0}\right]{,}\left[{1}{,}{1}{,}{2}{,}{1}\right]{,}\left[{0}{,}{0}{,}{1}{,}{0}\right]{,}\left[{1}{,}{0}{,}{1}{,}{1}\right]{,}\left[{0}{,}{0}{,}{0}{,}{1}\right]\right]$ (2.6)