ParabolicSubalgebra - Maple Help

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

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

Example 1.

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

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

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

 sl4 > $P≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sl4}\right):$

Here are the properties we need:

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

The possible subsets of the simple roots are:

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

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

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

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

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

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

 sl4 > $\mathrm{ParabolicSubalgebraRoots}\left(\mathrm{PS7},P\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]$ (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 > $\mathrm{LD2}≔\mathrm{SimpleLieAlgebraData}\left("so\left(5,3\right)",\mathrm{so53},\mathrm{labelformat}="gl",\mathrm{labels}=\left[R,\mathrm{\theta }\right]\right):$
 sl4 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: so53}}$ (2.18)

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

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

The possible subsets of restricted simple roots are:

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

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

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



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

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

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

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