AreCommuting - Maple Programming Help

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AreCommuting

check if one LAVF commutes with another

 Calling Sequence AreCommuting(L, M) AreCommuting(L, M, N)

Parameters

 L, M, N - LAVF objects.

Description

 • 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.

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)

We can check if L is closed under commutator ...

 > $\mathrm{AreCommuting}\left(L,L,L\right)$
 ${\mathrm{true}}$ (4)

or by using a more direct call.

 > $\mathrm{IsLieAlgebra}\left(L\right)$
 ${\mathrm{true}}$ (5)

We can also check if L is abelian...

 > $\mathrm{AreCommuting}\left(L,L\right)$
 ${\mathrm{false}}$ (6)

or by using a more direct call.

 > $\mathrm{IsAbelian}\left(L\right)$
 ${\mathrm{false}}$ (7)

As we know the centre of L must be abelian,

 > $\mathrm{AreCommuting}\left(\mathrm{Centre}\left(L\right),\mathrm{Centre}\left(L\right)\right)$
 ${\mathrm{true}}$ (8)

Compatibility

 • The AreCommuting command was introduced in Maple 2020.