IsNilpotent - Maple Help

calculate the nilradical of of a LAVF object.

LowerCentralSeries

calculate the lower central series of a LAVF object.

UpperCentralSeries

calculate the upper central series of a LAVF object.

Hypercentre

calculate the hypercentre of a LAVF object.

IsNilpotent

check if a LAVF object is nilpotent.

 Calling Sequence Nilradical( obj) NilRadical( obj) LowerCentralSeries( obj) UpperCentralSeries( obj) Hypercentre( obj) Hypercenter( obj) IsNilpotent( obj)

Parameters

 obj - a LAVF object that is a Lie algebra i.e.IsLieAlgebra(obj) returns true, see IsLieAlgebra.

Description

 • Let L be a LAVF object which is a Lie algebra. Then the Nilradical method returns the nilradical of L (i.e. its largest nilpotent ideal), as a LAVF object.
 • The name NilRadical is provided as an alias.
 • Let L be a LAVF object which is a Lie algebra. Then LowerCentralSeries(L) returns the lower central series of L, as a list of LAVF objects.
 • By definition, the lower central series of L is the sequence of ideals $L={L}_{\left(1\right)}\supset {L}_{\left(2\right)}\supset \cdots \supset {L}_{\left(i\right)}\supset \cdots \supset {L}_{\left(k\right)}$ where ${L}_{\left(i+1\right)}≔\left[L,{L}_{\left(i\right)}\right]\cdot$
 • Similarly, the call UpperCentralSeries(L) returns the upper central series of L, as a list of LAVF objects.
 • Let L be a LAVF object which is a Lie algebra. Then Hypercentre(L) returns the hypercentre of L (i.e. last term of the upper central series), as a LAVF object.
 • The name Hypercenter is provided as an alias.
 • The call IsNilpotent(L) returns true if and only if the last term of the lower central series of L is trivial (i.e. ${L}_{k}=0$).
 • These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

Construct a LAVF for E(2).

 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)
 > $\mathrm{IsLieAlgebra}\left(L\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{Nilradical}\left(L\right)$
 $\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (5)
 > $\mathrm{UCS}≔\mathrm{UpperCentralSeries}\left(L\right)$
 ${\mathrm{UCS}}{≔}\left[\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{0}{,}{\mathrm{\eta }}{=}{0}\right]\right\}\right]$ (6)
 > $\mathrm{LCS}≔\mathrm{LowerCentralSeries}\left(L\right)$
 ${\mathrm{LCS}}{≔}\left[\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}{,}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}\right]$ (7)

By definition, the last term of the upper central series should be identical to the hypercentre.

 > $\mathrm{Hypercentre}\left(L\right)$
 $\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{0}{,}{\mathrm{\eta }}{=}{0}\right]\right\}$ (8)
 > $\mathrm{AreSame}\left(\mathrm{Hypercentre}\left(L\right),\mathrm{UCS}\left[-1\right]\right)$
 ${\mathrm{true}}$ (9)

The last term of the lower central series of L (LCS) is not trivial. Therefore, L is not nilpotent.

 > $\mathrm{IsNilpotent}\left(L\right)$
 ${\mathrm{false}}$ (10)
 > $\mathrm{AreSame}\left(\mathrm{Hypercentre}\left(L\right),L\right)$
 ${\mathrm{false}}$ (11)

Compatibility

 • The Nilradical, LowerCentralSeries, UpperCentralSeries, Hypercentre and IsNilpotent commands were introduced in Maple 2020.