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 and$\mathrm{\𝔤}\mathit{ }\mathit{\=}\mathit{ }\mathrm{\𝔥}\mathit{ }\oplus \left[\underset{\mathrm{\alpha } \in {\mathrm{\Delta }}^{\+}}{⨁}{R}_{\mathrm{\α}}\right]$the associated root space decomposition. Let ${\mathrm{\Δ}}^{\+}$be a choice of positive roots and let ${\mathrm{\Δ}}^0 \subseteq {\mathrm{\Δ}}^{\+}$be a set of simple roots. The subalgebra $\mathrm{\𝔟}\mathit{ }\mathit{\=}\mathit{ }\mathrm{\𝔥}\mathit{ }\oplus \left[\underset{\mathrm{\α} \in {\mathrm{\Delta }}^{}}{⨁}{R}_{\mathrm{\alpha } }\right]$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.)$\mathrm{\alpha } \in {\mathrm{\Delta }}^{\+}$

 Given a standard parabolic subalgebra p , let ${\mathrm{\Φ}}_{{}^{}\mathrm{\𝔭} }^0 \= \{\mathrm{\α} \in {\mathrm{\Δ}}^0 \| R_{-\mathrm{\α}} \subseteq \mathrm{\𝔭} \}\.$This set of simple roots completely specifies the parabolic subalgebra p. Conversely, given a set of simple roots ${\mathrm{\Φ}}^0\,$let $\mathrm{\Φ} \=\{ \mathrm{\alpha } \in {\mathrm{\Delta }}^{\+} \| {\mathrm{\α}}^{}$is a linear combination of the roots in ${\mathrm{\Φ}}^0 \}$and set $\mathrm{\𝔭}{\mathit{ }}_{{\mathrm{\Φ}}^0}\mathit{\=}\mathit{ }\mathrm{\𝔥}\mathit{ }\oplus \left[\underset{\mathrm{\α} \in \mathrm{\Φ}}{⨁}{R}_{\mathrm{\alpha } } \right]$. Then ${\mathrm{\𝔭}}_{{\mathrm{\Φ}}^0}$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 $\mathrm{\𝔤}\mathit{ }\mathit{\=}\mathit{ }Z\left(\mathrm{\𝔞}\right)\oplus \left[\underset{\mathrm{\alpha } \in {\mathrm{\Delta }}^{}}{⨁}{S}_{\mathrm{\α}}\right]$relative to a maximal Abelian subalgebra a on which the Killing form is positive-definite.

Let Σ be a subset of the simple roots ${\mathrm{\Delta }}^0$and set ${\mathrm{\Phi }}^0$ = ${\mathrm{\Delta }}^0\/{\mathrm{\Σ}}_{}\.$ The command ParabolicSubalgebra returns the standard parabolic subalgebra ${\mathrm{\𝔭}}_{{\mathrm{\Φ}}^0}\.$The command ParabolicSubalgebraRoots returns the list of simple roots ${\mathrm{\Σ}}_{} \.$

With the keyword argument method = "non-compact", a real parabolic subalgebra is calculated.

With ${\mathrm{\Σ}}_{} \= {\mathrm{\Δ}}^0\,$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 $\mathrm{T1}$or $\mathrm{T2}\.$ 

The command Query/"Parabolic" will test if a given subalgebra of a semi-simple Lie algebra is parabolic.

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

$\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left(''sl(4)''\,\mathrm{sl4}\,\mathrm{labelformat}=''gl''\,\mathrm{labels}=\left[E\,\mathrm{\omega }\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; sl4}$

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

$\mathrm{CSA}≔P\left[''CartanSubalgebra''\right]$

$\mathrm{CSA}:=\left[\mathrm{E11}\,\mathrm{E22}\,\mathrm{E33}\right]$

$\mathrm{RSD}≔\mathrm{eval}\left(P\left[''RootSpaceDecomposition''\right]\right)$

$\mathrm{RSD}:=\mathrm{table}\left(\left[\left[1\,-1\,0\right]\=\mathrm{E12}\,\left[-1\,1\,0\right]\=\mathrm{E21}\,\left[-2\,-1\,-1\right]\=\mathrm{E41}\,\left[1\,2\,1\right]\=\mathrm{E24}\,\left[1\,0\,-1\right]\=\mathrm{E13}\,\left[-1\,-1\,-2\right]\=\mathrm{E43}\,\left[0\,1\,-1\right]\=\mathrm{E23}\,\left[1\,1\,2\right]\=\mathrm{E34}\,\left[-1\,0\,1\right]\=\mathrm{E31}\,\left[-1\,-2\,-1\right]\=\mathrm{E42}\,\left[2\,1\,1\right]\=\mathrm{E14}\,\left[0\,-1\,1\right]\=\mathrm{E32}\right]\right)$

$\mathrm{SR}≔P\left[''SimpleRoots''\right]$

$\mathrm{PR}≔P\left[''PositiveRoots''\right]$

$\mathrm{\Sigma }≔\left[\left[\right]\,\mathrm{SR}\left[1..1\right]\,\mathrm{SR}\left[2..2\right]\,\mathrm{SR}\left[3..3\right]\,\mathrm{SR}\left[1..2\right]\,\mathrm{SR}\left[2..3\right]\,\left[\mathrm{SR}\left[1\right]\,\mathrm{SR}\left[3\right]\right]\,\mathrm{SR}\right]$

$\mathrm{\Sigma }\left[1\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[1\right]\,P\right)$

$\left[\right]\,\left[\mathrm{E11}\,\mathrm{E22}\,\mathrm{E33}\,\mathrm{E12}\,\mathrm{E13}\,\mathrm{E14}\,\mathrm{E21}\,\mathrm{E23}\,\mathrm{E24}\,\mathrm{E31}\,\mathrm{E32}\,\mathrm{E34}\,\mathrm{E41}\,\mathrm{E42}\,\mathrm{E43}\right]$

$\mathrm{\Sigma }\left[2\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[2\right]\,P\right)$

$\mathrm{\Sigma }\left[3\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[3\right]\,P\right)$

$\mathrm{\Sigma }\left[4\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[4\right]\,P\right)$

$\mathrm{\Sigma }\left[5\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[5\right]\,P\right)$

$\mathrm{\Sigma }\left[6\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[6\right]\,P\right)$

$\mathrm{\Sigma }\left[7\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[7\right]\,P\right)$

$\mathrm{\Sigma }\left[8\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[8\right]\,P\right)$

$\mathrm{PS7}≔\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[7\right]\,P\right)$

$\mathrm{PS7}:=\left[\mathrm{E11}\,\mathrm{E22}\,\mathrm{E33}\,\mathrm{E12}\,\mathrm{E13}\,\mathrm{E14}\,\mathrm{E23}\,\mathrm{E24}\,\mathrm{E32}\,\mathrm{E34}\right]$

$\mathrm{Query}\left(\mathrm{PS7}\,''Parabolic''\right)$

$\mathrm{true}$

$\mathrm{ParabolicSubalgebraRoots}\left(\mathrm{PS7}\,P\right)$

$\mathrm{LD2}≔\mathrm{SimpleLieAlgebraData}\left(''so(5,3)''\,\mathrm{so53}\,\mathrm{labelformat}=''gl''\,\mathrm{labels}=\left[R\,\mathrm{\theta }\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{LD2}\right)$

$\mathrm{Lie\; algebra:\; so53}$

$P≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{so53}\right)\:$

$\mathrm{RRSD}≔\mathrm{eval}\left(P\left[''RestrictedRootSpaceDecomposition''\right]\right)$

$\mathrm{RRSD}:=\mathrm{table}\left(\left[\left[0\,0\,1\right]\=\left[\mathrm{R37}\,\mathrm{R38}\right]\,\left[1\,-1\,0\right]\=\left[\mathrm{R12}\right]\,\left[0\,-1\,0\right]\=\left[\mathrm{R57}\,\mathrm{R58}\right]\,\left[1\,0\,1\right]\=\left[\mathrm{R16}\right]\,\left[-1\,1\,0\right]\=\left[\mathrm{R21}\right]\,\left[0\,0\,-1\right]\=\left[\mathrm{R67}\,\mathrm{R68}\right]\,\left[1\,1\,0\right]\=\left[\mathrm{R15}\right]\,\left[0\,1\,1\right]\=\left[\mathrm{R26}\right]\,\left[-1\,0\,0\right]\=\left[\mathrm{R47}\,\mathrm{R48}\right]\,\left[1\,0\,-1\right]\=\left[\mathrm{R13}\right]\,\left[1\,0\,0\right]\=\left[\mathrm{R17}\,\mathrm{R18}\right]\,\left[-1\,-1\,0\right]\=\left[\mathrm{R42}\right]\,\left[0\,1\,-1\right]\=\left[\mathrm{R23}\right]\,\left[-1\,0\,-1\right]\=\left[\mathrm{R43}\right]\,\left[0\,1\,0\right]\=\left[\mathrm{R27}\,\mathrm{R28}\right]\,\left[-1\,0\,1\right]\=\left[\mathrm{R31}\right]\,\left[0\,-1\,1\right]\=\left[\mathrm{R32}\right]\,\left[0\,-1\,-1\right]\=\left[\mathrm{R53}\right]\right]\right)$

$\mathrm{RSR}≔P\left[''RestrictedSimpleRoots''\right]$

$\mathrm{\Sigma }≔\left[\mathrm{RSR}\,\mathrm{RSR}\left[1..2\right]\,\mathrm{RSR}\left[2..3\right]\,\left[\mathrm{RSR}\left[1\right]\,\mathrm{RSR}\left[3\right]\right]\,\mathrm{RSR}\left[1..1\right]\,\mathrm{RSR}\left[2..2\right]\,\mathrm{RSR}\left[3..3\right]\,\left[\right]\right]$

$\mathrm{\Sigma }\left[1\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[1\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{\Sigma }\left[2\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[2\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{\Sigma }\left[3\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[3\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{\Sigma }\left[4\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[4\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{\Sigma }\left[5\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[5\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{\Sigma }\left[6\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[6\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{\Sigma }\left[7\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[7\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{\Sigma }\left[8\right],\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[8\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\left[\right]\,\left[\mathrm{R11}\,\mathrm{R12}\,\mathrm{R13}\,\mathrm{R21}\,\mathrm{R22}\,\mathrm{R23}\,\mathrm{R31}\,\mathrm{R32}\,\mathrm{R33}\,\mathrm{R15}\,\mathrm{R16}\,\mathrm{R26}\,\mathrm{R42}\,\mathrm{R43}\,\mathrm{R53}\,\mathrm{R17}\,\mathrm{R18}\,\mathrm{R27}\,\mathrm{R28}\,\mathrm{R37}\,\mathrm{R38}\,\mathrm{R47}\,\mathrm{R48}\,\mathrm{R57}\,\mathrm{R58}\,\mathrm{R67}\,\mathrm{R68}\,\mathrm{R78}\right]$

$\mathrm{PS5}≔\mathrm{ParabolicSubalgebra}\left(\mathrm{\Sigma }\left[5\right]\,P\,\mathrm{method}=''non-compact''\right)$

$\mathrm{PS5}:=\left[\mathrm{R11}\,\mathrm{R12}\,\mathrm{R13}\,\mathrm{R22}\,\mathrm{R23}\,\mathrm{R32}\,\mathrm{R33}\,\mathrm{R15}\,\mathrm{R16}\,\mathrm{R26}\,\mathrm{R53}\,\mathrm{R17}\,\mathrm{R18}\,\mathrm{R27}\,\mathrm{R28}\,\mathrm{R37}\,\mathrm{R38}\,\mathrm{R57}\,\mathrm{R58}\,\mathrm{R67}\,\mathrm{R68}\,\mathrm{R78}\right]$

$\mathrm{Query}\left(\mathrm{PS5}\,''Parabolic''\right)$

$\mathrm{true}$

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