Let M, L be LAVF objects which are Lie algebras (see IsLieAlgebra) and  $M\subseteq L$  . Then Centraliser(M, L) computes the centraliser of M in L, namely $\left\{x\in L\|\left[x\,M\right]\=0\right\}$, as a new LAVF object.

Centraliser(M, L) is equivalent to Transporter(L, M, T) where T is a trivial LAVF (i.e. its determining system has trivial solutions) associated with vector fields of L. See the method Transporter for more detail.

The name Centralizer is provided as alias.

Similarly, let L, U be LAVF objects that are Lie algebras and $L\subseteq U$. Then Normaliser(L,U) computes the normaliser of L in U, namely $\left\{x\in U\|\left[x\,L\right]\in L\right\}$ , as a new LAVF object.

The second input argument U defaults to a universal LAVF associated with vector fields of L. That is, Normaliser(L) is equivalent to Normaliser(L, U) where U := LAVF(GetVectorField(L), "universal").

The call Normaliser(L,U) is equivalent to Transporter(U, L, L).

These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

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

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

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

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

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

$\mathrm{T2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{T2}≔\left[{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

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

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

$\mathrm{LT2}≔\mathrm{LAVF}\left(V\,\mathrm{T2}\right)$

$\mathrm{LT2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{LE2}≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$\mathrm{LE2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{IsLieAlgebra}\left(\mathrm{LT2}\right)$

$\mathrm{true}$

$\mathrm{IsLieAlgebra}\left(\mathrm{LE2}\right)$

$\mathrm{true}$

$\mathrm{IsSubspace}\left(\mathrm{LT2}\,\mathrm{LE2}\right)$

$\mathrm{true}$

$\mathrm{Centraliser}\left(\mathrm{LT2}\,\mathrm{LE2}\right)$

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

$\mathrm{Normaliser}\left(\mathrm{LT2}\,\mathrm{LE2}\right)$

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

$\mathrm{N2}≔\mathrm{Normaliser}\left(\mathrm{LE2}\right)$

$\mathrm{N2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_{y,y}=0\,{\mathrm{\xi }}_x={\mathrm{\eta }}_y\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\right]\right\}$

$\mathrm{SolutionDimension}\left(\mathrm{N2}\right)$

$4$

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

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

$L≔\mathrm{LAVF}\left(V\,S\right)$

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

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

$\mathrm{true}$

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

$2$

$N≔\mathrm{Normaliser}\left(L\right)$

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

$\mathrm{SolutionDimension}\left(N\right)$

$\mathrm{\infty }$

The Centraliser and Normaliser commands were introduced in Maple 2020.

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