Let $\mathrm{\𝔤}\mathit{ }$be a Lie algebra, $S \subset \mathrm{\𝔤}\mathit{ }$a subalgebra and $M \subset \mathrm{\𝔤}\mathit{ }$a subspace. Then the subalgebra, subspace pair $S\, M$is called a reductive pair if [i] $\mathrm{\𝔤}\mathit{ }\mathit{\=}\mathit{ }S\mathit{ }\mathit{\oplus }\mathit{ }M$(vector space direct sum) and [ii] $\left[x\,y\right] \in M$ for all $x \in S$and y in $M$. The subspace$M$ is called a reductive complement for the subalgebra $S$.

Query(S, M, "ReductivePair") returns true if the subspace M defines a reductive complement to the subalgebra S.

Query(S, M, parm, "ReductivePair") returns a sequence TF, Eq, Soln, reductiveList. Here TF is true if Maple finds parameter values for which M is a reductive complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for M to be a reductive complement; Soln is the list of solutions to the equations Eq; and reductiveList is the list of reductive subspaces obtained from the parameter values given by the different solutions in Soln.

The command Query is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).

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

$L≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,1\right]\,0\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,4\,2\right]\,1\right]\,\left[\left[3\,4\,3\right]\,-1\right]\right]\right]\right)$

$L:=\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=-\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(L\right)\:$

$\mathrm{S1}≔\left[\mathrm{e1}\,\mathrm{e2}\right]\:$$\mathrm{M1}≔\left[\mathrm{e3}\,\mathrm{e4}\right]\:$

$\mathrm{Query}\left(\mathrm{S1}\,\mathrm{M1}\,''ReductivePair''\right)$

$\mathrm{false}$

$\mathrm{S2}≔\left[\mathrm{e3}\,\mathrm{e4}\right]\:$$\mathrm{M2}≔\left[\mathrm{e1}\,\mathrm{e2}\right]\:$

$\mathrm{Query}\left(\mathrm{S2}\,\mathrm{M2}\,''ReductivePair''\right)$

$\mathrm{true}$

$\mathrm{S3}≔\left[\mathrm{e3}\,\mathrm{e4}\right]\:$$\mathrm{M3}≔\mathrm{evalDG}\left(\left[\mathrm{e1}+\mathrm{a1}\mathrm{e3}+\mathrm{a2}\mathrm{e4}\,\mathrm{e2}+\mathrm{a3}\mathrm{e3}+\mathrm{a4}\mathrm{e4}\right]\right)\:$

$\mathrm{TF},\mathrm{EQ},\mathrm{SOL},\mathrm{redPair}≔\mathrm{Query}\left(\mathrm{S3}\,\mathrm{M3}\,\left\{\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\,\mathrm{a4}\right\}\,''ReductivePair''\right)$

$\mathrm{TF}\,\mathrm{EQ}\,\mathrm{SOL}\,\mathrm{redPair}:=\mathrm{true}\,\left\{0\,\mathrm{a2}\,-\mathrm{a1}\,-\mathrm{a2}\,-2\mathrm{a3}\,-\mathrm{a4}\,-\mathrm{a1}\+\mathrm{a4}\right\}\,\left[\left\{\mathrm{a1}\=0\,\mathrm{a2}\=0\,\mathrm{a3}\=0\,\mathrm{a4}\=0\right\}\right]\,\left[\left[\left[\mathrm{e3}\,\mathrm{e4}\right]\,\left[\mathrm{e1}\,\mathrm{e2}\right]\right]\right]$

$\mathrm{redPair}\left[1\right]$

$\left[\left[\mathrm{e3}\,\mathrm{e4}\right]\,\left[\mathrm{e1}\,\mathrm{e2}\right]\right]$

$\mathrm{S4}≔\left[\mathrm{e4}\right]\:$

$\mathrm{M4}≔\mathrm{ComplementaryBasis}\left(\mathrm{S4}\,\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]\,a\right)$

$\mathrm{M4}:=\left[\mathrm{e1}\+\mathrm{a1}\mathrm{e4}\,\mathrm{e2}\+\mathrm{a2}\mathrm{e4}\,\mathrm{e3}\+\mathrm{a3}\mathrm{e4}\right]\,\left\{\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\right\}$

$\mathrm{Query}\left(\mathrm{S4}\,\mathrm{M4}\,''ReductivePair''\right)$

$\mathrm{true}\,\left\{0\,\mathrm{a3}\,-\mathrm{a2}\right\}\,\left[\left\{\mathrm{a1}\=\mathrm{a1}\,\mathrm{a2}\=0\,\mathrm{a3}\=0\right\}\right]\,\left[\left[\left[\mathrm{e4}\right]\,\left[\mathrm{e1}\+\mathrm{a1}\mathrm{e4}\,\mathrm{e2}\,\mathrm{e3}\right]\right]\right]$