DifferentialGeometry/LieAlgebras/Query/ParabolicSubalgebra - Maple Help

Query[ParabolicSubalgebra] - check if a list of vectors defines a parabolic subalgebra of a semi-simple Lie algebra

Calling Sequences

Query()

Parameters

P     - a list of vectors, defining a subalgebra of a semi-simple Lie algebra

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. Alternatively, a subalgebra p is parabolic if its nilradical is the orthogonal complement of p with respect to the Killing form $B.$
 • This Query command returns true if the subalgebra p defined by the vectors $P$ satisfies

Examples

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

We check to see if 3 subalgebras of $\mathrm{sl}\left(3\right)$are parabolic.  We construct the Lie algebra $\mathrm{sl}\left(3\right)$directly from its standard matrix representation.

 > $M≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[1,0,0\right],\left[0,0,0\right],\left[0,0,-1\right]\right],\left[\left[0,0,0\right],\left[0,1,0\right],\left[0,0,-1\right]\right],\left[\left[0,1,0\right],\left[0,0,0\right],\left[0,0,0\right]\right],\left[\left[0,0,1\right],\left[0,0,0\right],\left[0,0,0\right]\right],\left[\left[0,0,0\right],\left[1,0,0\right],\left[0,0,0\right]\right],\left[\left[0,0,0\right],\left[0,0,1\right],\left[0,0,0\right]\right],\left[\left[0,0,0\right],\left[0,0,0\right],\left[1,0,0\right]\right],\left[\left[0,0,0\right],\left[0,0,0\right],\left[0,1,0\right]\right]\right]\right)$
 ${M}{:=}\left[\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {1}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {1}& {0}\end{array}\right]\right]$ (2.1)
 > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(M,\mathrm{sl3}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{+}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]$ (2.2)

Initialize the Lie algebra. We label the basis elements for in a manner consistent with its matrix representation.

 > $\mathrm{DGsetup}\left(\mathrm{LD},\left[\mathrm{E11},\mathrm{E22},\mathrm{E12},\mathrm{E13},\mathrm{E21},\mathrm{E23},\mathrm{E31},\mathrm{E32}\right],\left[\mathrm{ω11},\mathrm{ω22},\mathrm{ω12},\mathrm{ω13},\mathrm{ω21},\mathrm{ω23},\mathrm{ω31},\mathrm{ω32}\right]\right)$
 ${\mathrm{Lie algebra: sl3}}$ (2.3)

Subalgebra 1.

 sl3 > $\mathrm{P1}≔\left[\mathrm{E11},\mathrm{E22},\mathrm{E12},\mathrm{E23},\mathrm{E13},\mathrm{E21}\right]$
 ${\mathrm{P1}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E13}}{,}{\mathrm{E21}}\right]$ (2.4)
 sl3 > $\mathrm{Query}\left(\mathrm{P1},"Parabolic"\right)$
 ${\mathrm{true}}$ (2.5)

Subalgebra 2.

 sl3 > $\mathrm{P2}≔\left[\mathrm{E11},\mathrm{E22},\mathrm{E12},\mathrm{E23},\mathrm{E13},\mathrm{E32}\right]$
 ${\mathrm{P2}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E13}}{,}{\mathrm{E32}}\right]$ (2.6)
 sl3 > $\mathrm{Query}\left(\mathrm{P2},"Parabolic"\right)$
 ${\mathrm{true}}$ (2.7)

Subalgebra 3.

 sl3 > $\mathrm{P3}≔\left[\mathrm{E11},\mathrm{E32},\mathrm{E23},\mathrm{E22}\right]$
 ${\mathrm{P3}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E32}}{,}{\mathrm{E23}}{,}{\mathrm{E22}}\right]$ (2.8)
 sl3 > $\mathrm{Query}\left(\mathrm{P3},"Subalgebra"\right)$
 ${\mathrm{true}}$ (2.9)
 sl3 > $\mathrm{Query}\left(\mathrm{P3},"Parabolic"\right)$
 ${\mathrm{false}}$ (2.10)