SimpleRoots - Maple 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

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.

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

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

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

Here is the corresponding root space decomposition.

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

We calculate the positive roots for .

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

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

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

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

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

 See Also