 DifferentialGeometry/LieAlgebras/Query/Ideal - Maple Help

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Query[Ideal] - check if a subalgebra defines an ideal in a Lie algebra

Calling Sequences

Query(S, "Ideal")

Query(S, parm, "Ideal")

Parameters

S       -  a list of independent vectors which defines a basis for subalgebra in a Lie algebra

parm    - (optional) a set of parameters appearing in the list of vectors S; it is assumed that the set of vectors S is well-defined when the parameters vanish Description

 • A list of vectors  in a Lie algebra is a basis for an ideal in  if span(for all and .
 • Query(S, "Ideal") returns true if the subalgebra S defines an ideal and false otherwise.
 • Query(S, parm, "Ideal") returns a sequence TF, Eq, Soln, IdealList.  Here TF is true if Maple finds parameter values for which S is an ideal and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S to be an ideal; Soln is the list of solutions to the equations Eq; and IdealList is the list of ideals 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(...). Examples

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

Example 1.

First initialize a Lie algebra; then define some subalgebras  and check to see if they are ideals.

 Alg > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg},\left[4\right]\right],\left[\left[\left[1,4,1\right],0\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right],\left[\left[3,4,3\right],-1\right]\right]\right]\right)$
 ${L}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}\right]$ (2.1)
 Alg > $\mathrm{DGsetup}\left(L\right):$
 Alg > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e2}\right]$
 ${\mathrm{S1}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]$ (2.2)
 Alg > $\mathrm{Query}\left(\mathrm{S1},"Ideal"\right)$
 ${\mathrm{true}}$ (2.3)
 Alg > $\mathrm{S2}≔\left[\mathrm{e3},\mathrm{e4}\right]$
 ${\mathrm{S2}}{:=}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]$ (2.4)
 Alg > $\mathrm{Query}\left(\mathrm{S2},"Ideal"\right)$
 ${\mathrm{false}}$ (2.5)

The subalgebra ${S}_{3}$depends on a parameter ${a}_{1}$.  We find which parameter values make ${S}_{3}$ an ideal.

 Alg > $\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{e2},\mathrm{e1}+\mathrm{a1}\mathrm{e4}\right]\right):$
 Alg > $\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},\mathrm{IdealList}≔\mathrm{Query}\left(\mathrm{S3},\left\{\mathrm{a1},\mathrm{a2}\right\},"Ideal"\right)$
 ${\mathrm{TF}}{,}{\mathrm{EQ}}{,}{\mathrm{SOLN}}{,}{\mathrm{IdealList}}{≔}{\mathrm{true}}{,}\left\{{0}{,}{-}{\mathrm{a1}}{,}{\mathrm{a1}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a2}}{=}{\mathrm{a2}}\right\}\right]{,}\left[\left[{\mathrm{e2}}{,}{\mathrm{e1}}\right]\right]$ (2.6)

The following equations must hold for to be an ideal (each expression must vanish).

 Alg > $\mathrm{EQ}$
 $\left\{{0}{,}{-}{\mathrm{a1}}{,}{\mathrm{a1}}\right\}$ (2.7)
 Alg > $\mathrm{S4}≔\mathrm{IdealList}\left[1\right]$
 ${\mathrm{S4}}{≔}\left[{\mathrm{e2}}{,}{\mathrm{e1}}\right]$ (2.8)
 Alg > $\mathrm{Query}\left(\mathrm{S4},"Ideal"\right)$
 ${\mathrm{true}}$ (2.9)