In the first calling sequence, AdjointMatrix(L) returns the adjoint representation matrix of L.

For AdjointMatrix(L) to make sense, the LAVF object L must be a Lie algebra (i.e. IsLieAlgebra(L) returns true. See IsLieAlgebra for more detail).

In the second calling sequence,  AdjointMatrix(L,M) returns a matrix representation of the Lie algebra L on the invariant subspace M. (i.e IsInvariant(M,L) returns true. See IsInvariant for more detail).

The third calling sequence is the general form of the method. AdjointMatrix(L,M,N) returns a matrix representing the action of L on M in N.

For AdjointMatrix(L, M, N) to make sense, L must commute with M modulo N (i.e. AreCommuting(L,M,N) returns true. See AreCommuting for more detail).

By specifying output = "basis", the output will be returned in terms of basis.

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{true}$

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

$\left[\begin{array}{ccc}0& {\mathrm{\xi }}_y& -\mathrm{\xi }\\ -{\mathrm{\xi }}_y& 0& \mathrm{\eta }\\ 0& 0& 0\end{array}\right]$

$\mathrm{AdjointMatrix}\left(L\,'\mathrm{output}'=''basis''\right)$

$\left[\left[\begin{array}{ccc}0& 1& 0\\ \mathrm{-1}& 0& 0\\ 0& 0& 0\end{array}\right]\,\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]\,\left[\begin{array}{ccc}0& 0& \mathrm{-1}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\right]$

The AdjointMatrix command was introduced in Maple 2020.

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