Let L, M, N be LAVF objects on the same space with same local coordinates. Then AreCommuting(L, M) checks if L commutes with M, i.e. if $\left[L\,M\right]\=0$.

Similarly, the three arguments call AreCommuting(L, M, N) checks if L commutes with M mod N, i.e. if $\left[L\,M\right]$ is in $N$.

This method is symmetric in the first two input arguments, that is, AreCommuting(L,M, N) is same as AreCommuting(M,L,N).

Some Lie algebraic methods (IsLieAlgebra, IsAbelian, and IsIdeal) are front-ends to AreCommuting.

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

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

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

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

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

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

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

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

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

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

$\mathrm{AreCommuting}\left(L\,L\,L\right)$

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{AreCommuting}\left(L\,L\right)$

$\mathrm{false}$

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

$\mathrm{false}$

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

$\mathrm{true}$

The AreCommuting command was introduced in Maple 2020.

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