Let L, M, N be LAVF objects living on the same space. Then Transporter(L,M,N) finds the transporter of M to N in L, as a new LAVF object.

By definition, the transporter of $M$ to $N$ in $L$consists of the vector fields in $L$ that map vector fields in $M$ to vector fields in $N$, under commutator. That is, it is the subspace $\left\{X_L\in L\|\left[X_L\,X_M\right]\in N\,\forall X_M\in M\right\}$ where  $L\,M\,N$ are subspaces of some Lie algebra.

Some Lie algebraic structural methods (Center, Centraliser, Normaliser, and UpperCentralSeries) are front-ends to Transporter.

This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

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

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

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

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

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

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

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

$\mathrm{S1}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{\eta }\left(x\,y\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{S1}≔\left[{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=0\,\mathrm{\eta }=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{S2}≔\mathrm{LHPDE}\left(\left[\mathrm{\xi }\left(x\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=\frac{\mathrm{\eta }\left(x\,y\right)}{x}\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{S2}≔\left[\mathrm{\xi }=0\,{\mathrm{\eta }}_x=\frac{\mathrm{\eta }}{x}\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$L≔\mathrm{LAVF}\left(V\,S\right)$

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

$\mathrm{L1}≔\mathrm{LAVF}\left(V\,\mathrm{S1}\right)$

$\mathrm{L1}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=0\,\mathrm{\eta }=0\right]\right\}$

$\mathrm{L2}≔\mathrm{LAVF}\left(V\,\mathrm{S2}\right)$

$\mathrm{L2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\eta }}_x=\frac{\mathrm{\eta }}{x}\,{\mathrm{\eta }}_y=0\,\mathrm{\xi }=0\right]\right\}$

$\mathrm{L0}≔\mathrm{LAVF}\left(V\,''trivial''\right)$

$\mathrm{L0}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\mathrm{\xi }=0\,\mathrm{\eta }=0\right]\right\}$

$\mathrm{Transporter}\left(L\,\mathrm{L2}\,\mathrm{L1}\right)$

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

$\mathrm{Transporter}\left(L\,L\,\mathrm{L0}\right)$

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

$\mathrm{Transporter}\left(L\,L\,\mathrm{Centre}\left(L\right)\right)$

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

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

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

The Transporter command was introduced in Maple 2020.

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