The Query function can be used in one of two ways to check various properties of a Lie algebra, subalgebra, or transformation.  In the first way the function simply returns true if the property defined by the keyword holds and false otherwise. In the second way, a set of parameters is specified and the function returns the following sequence - TF, Eq, Soln, A. Here TF is true if there is a choice of the parameters which makes the keyword property true; Eq is the list of equations which the parameters must satisfy for the property defined by the keyword to be be true; Soln is the list of all solutions to the equations as found by the Maple solve command; and A is the list of all algebraic structures defined by the previously listed solutions.

The argument keyword must be one of the following, entered as a string (in quotes    ):

Further information is available under ?Query[keyword], where keyword is from the above list.

A user can add new functionality to Query with the command Query:-addCheck(keyword, procedure).

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)\:$

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[4\right]\right]\,\left[\left[\left[1\,3\,1\right]\,1\right]\,\left[\left[2\,3\,2\right]\,1\right]\right]\right]\right)\:$

$\mathrm{L2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg2}\,\left[3\right]\right]\,\left[\left[\left[2\,3\,1\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[f\right]\,\left[\mathrm{\alpha }\right]\right)\:$

$\mathrm{Query}\left(\mathrm{Alg1}\,''Nilpotent''\right)$

$\mathrm{false}$

$\mathrm{Query}\left(\mathrm{Alg2}\,''Nilpotent''\right)$

$\mathrm{true}$

$\mathrm{Query}\left(\left[\mathrm{e1}\,\mathrm{e2}\right]\,''Ideal''\right)$

$\mathrm{true}$

$\mathrm{Query}\left(\left[\mathrm{e1}\,\mathrm{e2}+a\mathrm{e4}\right]\,\left\{a\right\}\,''Ideal''\right)$

$\mathrm{true},\left\{0\,-a\right\},\left[\left\{a=0\right\}\right],\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\right]$

$A≔\mathrm{Matrix}\left(\left[\left[\mathrm{a6}\,0\,0\,\mathrm{a1}\right]\,\left[0\,\mathrm{a5}\,0\,\mathrm{a2}\right]\,\left[0\,0\,\mathrm{a4}\,\mathrm{a3}\right]\right]\right)$

$A≔\left[\begin{array}{cccc}\mathrm{a6}& 0& 0& \mathrm{a1}\\ 0& \mathrm{a5}& 0& \mathrm{a2}\\ 0& 0& \mathrm{a4}& \mathrm{a3}\end{array}\right]$

$\mathrm{TF},\mathrm{Eq},\mathrm{Soln},B≔\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,A\,\left\{\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\,\mathrm{a4}\,\mathrm{a5}\,\mathrm{a6}\right\}\,''Homomorphism''\right)$

$\mathrm{TF},\mathrm{Eq},\mathrm{Soln},B≔\mathrm{true},\left\{0\,\mathrm{a5}\,\mathrm{a6}\,\mathrm{a4}\mathrm{a2}\,\mathrm{a5}\mathrm{a3}\,\mathrm{a5}\mathrm{a4}\,-\mathrm{a5}\,-\mathrm{a6}\,-\mathrm{a4}\mathrm{a2}\,-\mathrm{a5}\mathrm{a3}\,-\mathrm{a5}\mathrm{a4}\right\},\left[\left\{\mathrm{a1}=\mathrm{a1}\,\mathrm{a2}=\mathrm{a2}\,\mathrm{a3}=\mathrm{a3}\,\mathrm{a4}=0\,\mathrm{a5}=0\,\mathrm{a6}=0\right\}\,\left\{\mathrm{a1}=\mathrm{a1}\,\mathrm{a2}=0\,\mathrm{a3}=\mathrm{a3}\,\mathrm{a4}=\mathrm{a4}\,\mathrm{a5}=0\,\mathrm{a6}=0\right\}\right],\left[\left[\begin{array}{cccc}0& 0& 0& \mathrm{a1}\\ 0& 0& 0& \mathrm{a2}\\ 0& 0& 0& \mathrm{a3}\end{array}\right]\,\left[\begin{array}{cccc}0& 0& 0& \mathrm{a1}\\ 0& 0& 0& 0\\ 0& 0& \mathrm{a4}& \mathrm{a3}\end{array}\right]\right]$

f := proc() local C, k;
if nargs = 1 then ChangeLieAlgebraTo(args[1]) end if;
C := LieAlgebraSeries("Lower");
k := nops(C[3]);
if k = 0 then true else false end if;
end:

$\mathrm{Query}:-\mathrm{addCheck}\left(''two-step''\,f\right)$

$f$

$\mathrm{Query}\left(\mathrm{Alg1}\,''two-step''\right)$

$\mathrm{false}$

$\mathrm{Query}\left(\mathrm{Alg2}\,''two-step''\right)$

$\mathrm{false}$

$\mathrm{Query}\left(''Keywords''\right)$

$\left[''Abelian''\,''AbsolutelyIndecomposable''\,''AscendingIdealsBasis''\,''Associative''\,''CartanDecomposition''\,''CartanInvolution''\,''CartanSubalgebra''\,''ChevalleyBasis''\,''ClosedUnderConjugation''\,''ClosedUnderHermitianTransposition''\,''ClosedUnderTransposition''\,''Commutative''\,''CompactForm''\,''Derivation''\,''DirectSumDecomposition''\,''Filtration''\,''Gradation''\,''Homomorphism''\,''Ideal''\,''Indecomposable''\,''IntegerStructureConstants''\,''InvariantSubspace''\,''Jacobi''\,''Keywords''\,''LeviDecomposition''\,''MatrixAlgebra''\,''NaturallyReductivePair''\,''NilRepresentation''\,''Nilpotent''\,''Normalizer''\,''Parabolic''\,''ReductivePair''\,''RegularElement''\,''Representation''\,''RootSpaceDecomposition''\,''Semisimple''\,''SkewCommutative''\,''Solvable''\,''SolvableRepresentation''\,''SplitForm''\,''Subalgebra''\,''SymmetricPair''\,''two-step''\right]$