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.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$V≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$L≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$L≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{IsLieAlgebra}\left(L\right)$

$\mathrm{true}$

$\mathrm{Nilradical}\left(L\right)$

$\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{UCS}≔\mathrm{UpperCentralSeries}\left(L\right)$

$\mathrm{UCS}≔\left[\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\mathrm{\xi }=0\,\mathrm{\eta }=0\right]\right\}\right]$

$\mathrm{LCS}≔\mathrm{LowerCentralSeries}\left(L\right)$

$\mathrm{LCS}≔\left[\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}\,\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\right]\right\}\right]$

$\mathrm{Hypercentre}\left(L\right)$

$\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\mathrm{\xi }=0\,\mathrm{\eta }=0\right]\right\}$

$\mathrm{AreSame}\left(\mathrm{Hypercentre}\left(L\right)\,\mathrm{UCS}\left[-1\right]\right)$

$\mathrm{true}$

$\mathrm{IsNilpotent}\left(L\right)$

$\mathrm{false}$

$\mathrm{AreSame}\left(\mathrm{Hypercentre}\left(L\right)\,L\right)$

$\mathrm{false}$

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

For more information on Maple 2020 changes, see Updates in Maple 2020.