 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.

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

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

 sl4 > $Pâ‰”\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sl4}\right):$

Here are the properties we need:

 sl4 > $\mathrm{CSA}â‰”{P}_{"CartanSubalgebra"}$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}\right]$ (2.2)
 sl4 > $\mathrm{RSD}â‰”\mathrm{eval}\left({P}_{"RootSpaceDecomposition"}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{E12}}{,}\left[{-}{1}{,}{1}{,}{0}\right]{=}{\mathrm{E21}}{,}\left[{-}{2}{,}{-}{1}{,}{-}{1}\right]{=}{\mathrm{E41}}{,}\left[{1}{,}{2}{,}{1}\right]{=}{\mathrm{E24}}{,}\left[{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{E13}}{,}\left[{-}{1}{,}{-}{1}{,}{-}{2}\right]{=}{\mathrm{E43}}{,}\left[{0}{,}{1}{,}{-}{1}\right]{=}{\mathrm{E23}}{,}\left[{1}{,}{1}{,}{2}\right]{=}{\mathrm{E34}}{,}\left[{-}{1}{,}{0}{,}{1}\right]{=}{\mathrm{E31}}{,}\left[{-}{1}{,}{-}{2}{,}{-}{1}\right]{=}{\mathrm{E42}}{,}\left[{2}{,}{1}{,}{1}\right]{=}{\mathrm{E14}}{,}\left[{0}{,}{-}{1}{,}{1}\right]{=}{\mathrm{E32}}\right]\right)$ (2.3)
 sl4 > $\mathrm{SR}â‰”{P}_{"SimpleRoots"}$ sl4 > $\mathrm{PR}â‰”{P}_{"PositiveRoots"}$ The possible subsets of the simple roots are:

 sl4 > $\mathrm{Σ}â‰”\left[\left[\right],{\mathrm{SR}}_{1..1},{\mathrm{SR}}_{2..2},{\mathrm{SR}}_{3..3},{\mathrm{SR}}_{1..2},{\mathrm{SR}}_{2..3},\left[{\mathrm{SR}}_{1},{\mathrm{SR}}_{3}\right],\mathrm{SR}\right]$ The possible parabolic subalgebras of $\mathrm{sl}\left(4\right)$ are therefore:

 sl > ${\mathrm{Σ}}_{1},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{1},P\right)$
 $\left[{}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E31}}{,}{\mathrm{E32}}{,}{\mathrm{E34}}{,}{\mathrm{E41}}{,}{\mathrm{E42}}{,}{\mathrm{E43}}\right]$ (2.4)
 sl4 > ${\mathrm{Σ}}_{2},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{2},P\right)$ sl4 > ${\mathrm{Σ}}_{3},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{3},P\right)$ sl4 > ${\mathrm{Σ}}_{4},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{4},P\right)$ sl4 > ${\mathrm{Σ}}_{5},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{5},P\right)$ sl4 > ${\mathrm{Σ}}_{6},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{6},P\right)$ sl4 > ${\mathrm{Σ}}_{7},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{7},P\right)$ sl4 > ${\mathrm{Σ}}_{8},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{8},P\right)$ The Query command can be used to check that these subalgebras are parabolic subalgebra.

 sl4 > $\mathrm{PS7}â‰”\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{7},P\right)$
 ${\mathrm{PS7}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E32}}{,}{\mathrm{E34}}\right]$ (2.5)
 sl4 > $\mathrm{Query}\left(\mathrm{PS7},"Parabolic"\right)$
 ${\mathrm{true}}$ (2.6)

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

 sl4 > $\mathrm{ParabolicSubalgebraRoots}\left(\mathrm{PS7},P\right)$ 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 > $\mathrm{LD2}â‰”\mathrm{SimpleLieAlgebraData}\left("so\left(5,3\right)",\mathrm{so53},\mathrm{labelformat}="gl",\mathrm{labels}=\left[R,\mathrm{θ}\right]\right):$
 sl4 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: so53}}$ (2.7)

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

 so53 > $Pâ‰”\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{so53}\right):$
 so53 > $\mathrm{RRSD}â‰”\mathrm{eval}\left({P}_{"RestrictedRootSpaceDecomposition"}\right)$
 ${\mathrm{RRSD}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{0}{,}{1}\right]{=}\left[{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}\left[{1}{,}{-}{1}{,}{0}\right]{=}\left[{\mathrm{R12}}\right]{,}\left[{0}{,}{-}{1}{,}{0}\right]{=}\left[{\mathrm{R57}}{,}{\mathrm{R58}}\right]{,}\left[{1}{,}{0}{,}{1}\right]{=}\left[{\mathrm{R16}}\right]{,}\left[{-}{1}{,}{1}{,}{0}\right]{=}\left[{\mathrm{R21}}\right]{,}\left[{0}{,}{0}{,}{-}{1}\right]{=}\left[{\mathrm{R67}}{,}{\mathrm{R68}}\right]{,}\left[{1}{,}{1}{,}{0}\right]{=}\left[{\mathrm{R15}}\right]{,}\left[{0}{,}{1}{,}{1}\right]{=}\left[{\mathrm{R26}}\right]{,}\left[{-}{1}{,}{0}{,}{0}\right]{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}\right]{,}\left[{1}{,}{0}{,}{-}{1}\right]{=}\left[{\mathrm{R13}}\right]{,}\left[{1}{,}{0}{,}{0}\right]{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}\right]{,}\left[{-}{1}{,}{-}{1}{,}{0}\right]{=}\left[{\mathrm{R42}}\right]{,}\left[{0}{,}{1}{,}{-}{1}\right]{=}\left[{\mathrm{R23}}\right]{,}\left[{-}{1}{,}{0}{,}{-}{1}\right]{=}\left[{\mathrm{R43}}\right]{,}\left[{0}{,}{1}{,}{0}\right]{=}\left[{\mathrm{R27}}{,}{\mathrm{R28}}\right]{,}\left[{-}{1}{,}{0}{,}{1}\right]{=}\left[{\mathrm{R31}}\right]{,}\left[{0}{,}{-}{1}{,}{1}\right]{=}\left[{\mathrm{R32}}\right]{,}\left[{0}{,}{-}{1}{,}{-}{1}\right]{=}\left[{\mathrm{R53}}\right]\right]\right)$ (2.8)
 sl4 > $\mathrm{RSR}â‰”{P}_{"RestrictedSimpleRoots"}$ The possible subsets of restricted simple roots are:

 so53 > $\mathrm{Σ}â‰”\left[\mathrm{RSR},{\mathrm{RSR}}_{1..2},{\mathrm{RSR}}_{2..3},\left[{\mathrm{RSR}}_{1},{\mathrm{RSR}}_{3}\right],{\mathrm{RSR}}_{1..1},{\mathrm{RSR}}_{2..2},{\mathrm{RSR}}_{3..3},\left[\right]\right]$ The parabolic subalgebras defined by these sets of restricted roots are:

 so53 > ${\mathrm{Σ}}_{1},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{1},P,\mathrm{method}="non-compact"\right)$ so53 > ${\mathrm{Σ}}_{2},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{2},P,\mathrm{method}="non-compact"\right)$ so53 > ${\mathrm{Σ}}_{3},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{3},P,\mathrm{method}="non-compact"\right)$ so53 > ${\mathrm{Σ}}_{4},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{4},P,\mathrm{method}="non-compact"\right)$ so53 > ${\mathrm{Σ}}_{5},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{5},P,\mathrm{method}="non-compact"\right)$ so53 > ${\mathrm{Σ}}_{6},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{6},P,\mathrm{method}="non-compact"\right)$ so53 > ${\mathrm{Σ}}_{7},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{7},P,\mathrm{method}="non-compact"\right)$ so53 > ${\mathrm{Σ}}_{8},\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{8},P,\mathrm{method}="non-compact"\right)$
 $\left[{}\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R21}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R31}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R42}}{,}{\mathrm{R43}}{,}{\mathrm{R53}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}{,}{\mathrm{R78}}\right]$ (2.9)



Check that the subalgebra defined by  is parabolic.

 so53 > $\mathrm{PS5}â‰”\mathrm{ParabolicSubalgebra}\left({\mathrm{Σ}}_{5},P,\mathrm{method}="non-compact"\right)$
 ${\mathrm{PS5}}{:=}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R53}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}{,}{\mathrm{R78}}\right]$ (2.10)
 so53 > $\mathrm{Query}\left(\mathrm{PS5},"Parabolic"\right)$
 ${\mathrm{true}}$ (2.11)

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

 so53 > $\mathrm{ParabolicSubalgebraRoots}\left(\mathrm{PS5},P,\mathrm{method}="non-compact"\right)$ 