 Nilpotent - Maple Help

Query[Nilpotent] - check if a Lie algebra is nilpotent

Calling Sequences

Query(Alg, "Nilpotent")

Query(S, "Nilpotent")

Parameters

Alg     - (optional) the name of an initialized Lie algebra

S       - a list of vectors defining a basis for a subalgebra Description

 • A Lie algebra is nilpotent if the $k$-th ideal in the lower central series for  is 0 for some .
 • Query(Alg, "Nilpotent") returns true if Alg is a nilpotent Lie algebra and false otherwise.  If the algebra is unspecified, then Query is applied to the current algebra.
 • Query(S, "Nilpotent") returns true if the subalgebra spanned by the vectors  S is a nilpotent Lie algebra and false otherwise.
 • 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.

We initialize three different Lie algebras.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[3\right]\right],\left[\right]\right]\right)$
 ${\mathrm{L1}}{≔}\left[{}\right]$ (2.1)
 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[3\right]\right],\left[\left[\left[2,3,1\right],1\right]\right]\right]\right):$
 > $\mathrm{L3}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[4\right]\right],\left[\left[\left[1,4,1\right],2\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right],\left[\left[3,4,3\right],1\right]\right]\right]\right)$
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.2)
 > $\mathrm{DGsetup}\left(\mathrm{L1},\left[x\right],\left[a\right]\right):$$\mathrm{DGsetup}\left(\mathrm{L2},\left[y\right],\left[b\right]\right):$$\mathrm{DGsetup}\left(\mathrm{L3},\left[z\right],\left[c\right]\right):$

Alg1 and Alg2 are nilpotent but Alg3 is not.

 Alg3 > $\mathrm{Query}\left(\mathrm{Alg1},"Nilpotent"\right)$
 ${\mathrm{true}}$ (2.3)
 Alg1 > $\mathrm{Query}\left(\mathrm{Alg2},"Nilpotent"\right)$
 ${\mathrm{true}}$ (2.4)
 Alg2 > $\mathrm{Query}\left(\mathrm{Alg3},"Nilpotent"\right)$
 ${\mathrm{false}}$ (2.5)

The subalgebra  span of Alg3 is nilpotent but the subalgebra ${S}_{2}=$span is not.

 Alg3 > $\mathrm{S1}≔\left[\mathrm{z1},\mathrm{z2},\mathrm{z3}\right]:$$\mathrm{S2}≔\left[\mathrm{z1},\mathrm{z4}\right]:$
 Alg3 > $\mathrm{Query}\left(\mathrm{S1},"Nilpotent"\right)$
 ${\mathrm{true}}$ (2.6)
 Alg3 > $\mathrm{Query}\left(\mathrm{S2},"Nilpotent"\right)$
 ${\mathrm{false}}$ (2.7)