find the parabolic subalgebra defined by a set of simple roots or a set of restricted simple roots - Maple Programming Help

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LieAlgebras[ParabolicSubalgebra] - find the parabolic subalgebra defined by a set of simple roots or a set of restricted simple roots

LieAlgebras[ParabolicSubalgebraRoots] - find the simple roots which generate a parabolic subalgebra

Calling Sequences

ParabolicSubalgebra(${\mathbf{Σ}}$, T1)

ParabolicSubalgebra(, T2, method="non-compact")

ParabolicSubalgebraRoots(ParAlgT2)

ParabolicSubalgebraRoots(ParAlg, T2, method="non-compact")

Parameters

$\mathrm{Σ}$       - a list or set of column vectors, defining a subset of simple roots

T1      - a table, with indices that include "RootSpaceDecomposition", "CartanSubalgebra", "SimpleRoots", and "PositiveRoots"

T2      - a table, with indices that include "RestrictedRootSpaceDecomposition", "CartanSubalgebraDecomposition", "RestrictedSimpleRoots" and "RestrictedPositiveRoots"

ParAlg  - a list of vectors in a Lie algebra, defining a parabolic subalgebra

Description

 • Let g be a semi-simple Lie algebra. A Borel subalgebra  b is any maximal solvable subalgebra. A parabolic subalgebra p is any subalgebra containing a Borel subalgebra b. Alternatively, a subalgebra p is parabolic if its nilradical is the orthogonal complement of p with respect to the Killing form $B.$
 • Let h be an Cartan subalgebra andthe associated root space decomposition. Let be a choice of positive roots and let be a set of simple roots. The subalgebra is called the standard Borel subalgebra associated to h and any parabolic subalgebra containing it is called a standard parabolic subalgebra. (One could replace the summation over the positive roots by one over the negative roots.)
 • Given a standard parabolic subalgebra p , let This set of simple roots completely specifies the parabolic subalgebra p. Conversely, given a set of simple roots let is a linear combination of the roots in and set . Then is a standard parabolic subalgebra.
 • For the parabolic subalgebras of a real semi-simple Lie algebra the situation is essentially the same except that one must consider the restricted root space decomposition relative to a maximal Abelian subalgebra a on which the Killing form is positive-definite.
 • Let Σ be a subset of the simple roots and set ${\mathrm{\Phi }}^{0}$ = ${\mathrm{\Delta }}^{0}/{\mathrm{Σ}}_{}.$ The command ParabolicSubalgebra returns the standard parabolic subalgebra The command ParabolicSubalgebraRoots returns the list of simple roots
 • With the keyword argument method = "non-compact", a real parabolic subalgebra is calculated.
 • With the standard Borel subalgebra is returned.
 • If the Lie algebra is created from the command SimpleLieAlgebraData , then the table obtained from the command SimpleLieAlgebraProperties can be used as the second argument or $\mathrm{T2}.$
 • The command Query/"ParabolicSubalgebra" will test if a given subalgebra of a semi-simple Lie algebra is parabolic.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We calculate the parabolic subalgebras for We use the command SimpleLieAlgebraData to initialize the Lie algebra.

 > LD := SimpleLieAlgebraData("sl(4)", sl4, labelformat = "gl", labels = [E, omega]):
 > DGsetup(LD);
 ${\mathrm{Lie algebra: sl4}}$ (2.1)

We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, root space decomposition etc.

 sl4 > P := SimpleLieAlgebraProperties(sl4):

Here are the properties we need:

 sl4 > CSA := P["CartanSubalgebra"];
 ${\mathrm{CSA}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}\right]$ (2.2)
 sl4 > RSD := eval(P["RootSpaceDecomposition"]);
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{E12}}{,}\left[{-}{1}{,}{1}{,}{0}\right]{=}{\mathrm{E21}}{,}\left[{-}{2}{,}{-}{1}{,}{-}{1}\right]{=}{\mathrm{E41}}{,}\left[{1}{,}{2}{,}{1}\right]{=}{\mathrm{E24}}{,}\left[{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{E13}}{,}\left[{-}{1}{,}{-}{1}{,}{-}{2}\right]{=}{\mathrm{E43}}{,}\left[{0}{,}{1}{,}{-}{1}\right]{=}{\mathrm{E23}}{,}\left[{1}{,}{1}{,}{2}\right]{=}{\mathrm{E34}}{,}\left[{-}{1}{,}{0}{,}{1}\right]{=}{\mathrm{E31}}{,}\left[{-}{1}{,}{-}{2}{,}{-}{1}\right]{=}{\mathrm{E42}}{,}\left[{2}{,}{1}{,}{1}\right]{=}{\mathrm{E14}}{,}\left[{0}{,}{-}{1}{,}{1}\right]{=}{\mathrm{E32}}\right]\right)$ (2.3)
 sl4 > SR := P["SimpleRoots"];
 ${\mathrm{SR}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]$ (2.4)
 sl4 > PR := P["PositiveRoots"];
 ${\mathrm{PR}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {1}\\ {1}\end{array}\right]\right]$ (2.5)

The possible subsets of the simple roots are:

 sl4 > Sigma := [[], SR[1..1], SR[2..2], SR[3..3], SR[1..2], SR[2..3], [SR[1], SR[3]], SR];
 ${\mathrm{Σ}}{:=}\left[\left[{}\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]\right]$ (2.6)

The possible parabolic subalgebras of $\mathrm{sl}\left(4\right)$ are therefore:

 sl > Sigma[1], ParabolicSubalgebra(Sigma[1], P);
 $\left[{}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E31}}{,}{\mathrm{E32}}{,}{\mathrm{E34}}{,}{\mathrm{E41}}{,}{\mathrm{E42}}{,}{\mathrm{E43}}\right]$ (2.7)
 sl4 > Sigma[2], ParabolicSubalgebra(Sigma[2], P);
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E32}}{,}{\mathrm{E34}}{,}{\mathrm{E42}}{,}{\mathrm{E43}}\right]$ (2.8)
 sl4 > Sigma[3], ParabolicSubalgebra(Sigma[3], P);
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E34}}{,}{\mathrm{E43}}\right]$ (2.9)
 sl4 > Sigma[4], ParabolicSubalgebra(Sigma[4], P);
 $\left[\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E31}}{,}{\mathrm{E32}}{,}{\mathrm{E34}}\right]$ (2.10)
 sl4 > Sigma[5], ParabolicSubalgebra(Sigma[5], P);
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E34}}{,}{\mathrm{E43}}\right]$ (2.11)
 sl4 > Sigma[6], ParabolicSubalgebra(Sigma[6], P);
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E34}}\right]$ (2.12)
 sl4 > Sigma[7], ParabolicSubalgebra(Sigma[7], P);
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E32}}{,}{\mathrm{E34}}\right]$ (2.13)
 sl4 > Sigma[8], ParabolicSubalgebra(Sigma[8], P);
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E34}}\right]$ (2.14)

The Query command can be used to check that these subalgebras are parabolic subalgebra.

 sl4 > PS7 := ParabolicSubalgebra(Sigma[7], P);
 ${\mathrm{PS7}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E32}}{,}{\mathrm{E34}}\right]$ (2.15)
 sl4 > Query(PS7, "Parabolic");
 ${\mathrm{true}}$ (2.16)

With the command ParabolicSubalgebraRoots, we can find the simple roots used to create the parabolic algebra $\mathrm{PS7}$.

 sl4 > ParabolicSubalgebraRoots(PS7, P);
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]$ (2.17)

Example 2.

We calculate (real) parabolic subalgebras for $\mathrm{so}\left(6,3\right)$. We use the command SimpleLieAlgebraData to initialize the Lie algebra.

 sl4 > LD2 := SimpleLieAlgebraData("so(5,3)", so53, labelformat = "gl", labels = [R, theta]):
 sl4 > DGsetup(LD2);
 ${\mathrm{Lie algebra: so53}}$ (2.18)

We use the command SimpleLieAlgebraProperties to calculate the restricted root space decomposition and the restricted simple roots.

 so53 > P := SimpleLieAlgebraProperties(so53):
 so53 > RRSD := eval(P["RestrictedRootSpaceDecomposition"]);
 ${\mathrm{RRSD}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{0}{,}{1}\right]{=}\left[{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}\left[{1}{,}{-}{1}{,}{0}\right]{=}\left[{\mathrm{R12}}\right]{,}\left[{0}{,}{-}{1}{,}{0}\right]{=}\left[{\mathrm{R57}}{,}{\mathrm{R58}}\right]{,}\left[{1}{,}{0}{,}{1}\right]{=}\left[{\mathrm{R16}}\right]{,}\left[{-}{1}{,}{1}{,}{0}\right]{=}\left[{\mathrm{R21}}\right]{,}\left[{0}{,}{0}{,}{-}{1}\right]{=}\left[{\mathrm{R67}}{,}{\mathrm{R68}}\right]{,}\left[{1}{,}{1}{,}{0}\right]{=}\left[{\mathrm{R15}}\right]{,}\left[{0}{,}{1}{,}{1}\right]{=}\left[{\mathrm{R26}}\right]{,}\left[{-}{1}{,}{0}{,}{0}\right]{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}\right]{,}\left[{1}{,}{0}{,}{-}{1}\right]{=}\left[{\mathrm{R13}}\right]{,}\left[{1}{,}{0}{,}{0}\right]{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}\right]{,}\left[{-}{1}{,}{-}{1}{,}{0}\right]{=}\left[{\mathrm{R42}}\right]{,}\left[{0}{,}{1}{,}{-}{1}\right]{=}\left[{\mathrm{R23}}\right]{,}\left[{-}{1}{,}{0}{,}{-}{1}\right]{=}\left[{\mathrm{R43}}\right]{,}\left[{0}{,}{1}{,}{0}\right]{=}\left[{\mathrm{R27}}{,}{\mathrm{R28}}\right]{,}\left[{-}{1}{,}{0}{,}{1}\right]{=}\left[{\mathrm{R31}}\right]{,}\left[{0}{,}{-}{1}{,}{1}\right]{=}\left[{\mathrm{R32}}\right]{,}\left[{0}{,}{-}{1}{,}{-}{1}\right]{=}\left[{\mathrm{R53}}\right]\right]\right)$ (2.19)
 sl4 > RSR := P["RestrictedSimpleRoots"];
 ${\mathrm{RSR}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]$ (2.20)

The possible subsets of restricted simple roots are:

 so53 > Sigma := [RSR, RSR[1..2], RSR[2..3], [RSR[1], RSR[3]], RSR[1..1], RSR[2..2], RSR[3..3], []];
 ${\mathrm{Σ}}{:=}\left[\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[{}\right]\right]$ (2.21)

The parabolic subalgebras defined by these sets of restricted roots are:

 so53 > Sigma[1], ParabolicSubalgebra(Sigma[1], P, method = "non-compact");
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R78}}\right]$ (2.22)
 so53 > Sigma[2], ParabolicSubalgebra(Sigma[2], P, method = "non-compact");
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}{,}{\mathrm{R78}}\right]$ (2.23)
 so53 > Sigma[3], ParabolicSubalgebra(Sigma[3], P, method = "non-compact");
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R21}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R78}}\right]$ (2.24)
 so53 > Sigma[4], ParabolicSubalgebra(Sigma[4], P, method = "non-compact");
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R78}}\right]$ (2.25)
 so53 > Sigma[5], ParabolicSubalgebra(Sigma[5], P, method = "non-compact");
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R53}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}{,}{\mathrm{R78}}\right]$ (2.26)
 so53 > Sigma[6], ParabolicSubalgebra(Sigma[6], P, method = "non-compact");
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R21}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}{,}{\mathrm{R78}}\right]$ (2.27)
 so53 > Sigma[7], ParabolicSubalgebra(Sigma[7], P, method = "non-compact");
 $\left[\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R21}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R31}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R78}}\right]$ (2.28)
 so53 > Sigma[8], ParabolicSubalgebra(Sigma[8], P, method = "non-compact");
 $\left[{}\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R21}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R31}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R42}}{,}{\mathrm{R43}}{,}{\mathrm{R53}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}{,}{\mathrm{R78}}\right]$ (2.29)



Check that the subalgebra defined by (2.26) is parabolic.

 so53 > PS5 := ParabolicSubalgebra(Sigma[5], P, method = "non-compact");
 ${\mathrm{PS5}}{:=}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R53}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}{,}{\mathrm{R78}}\right]$ (2.30)
 so53 > Query(PS5, "Parabolic");
 ${\mathrm{true}}$ (2.31)

Find the restricted roots used to define $\mathrm{PS5}$ .

 so53 > ParabolicSubalgebraRoots(PS5, P, method = "non-compact");
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]$ (2.32)