Let  $\mathrm{\Γ}\= \left\{X_1\, X_2 \, ..\. X_p\right\}$ be a $p$-dimensional Lie algebra of vector fields, defined on a manifold $M\.$Let $\mathrm{\𝒢}\mathit{ }$be an infinite dimensional Lie algebra of vector fields on $M$, whose general element depends upon a certain number of arbitrary functions and suppose $\mathrm{\Γ} \subset \mathrm{\𝒢}\mathit{\.}$ Then the normalizer of $\mathrm{\Γ}$in $\mathrm{\𝒢}\mathit{ }$ is

 With the calling sequence InfinitesimalPseudoGroupNormalizer(Gamma), the normalizer of $\mathrm{\Γ}$ in the full infinitesimal pseudo-group of all vector fields on $M$ is computed. With output = "general" (the default value), a single vector field, depending upon arbitrary constants and functions, is returned. If the output depends only on constants (that is, Nor($$$\mathrm{\Γ}\, \mathrm{\𝒢}\mathit{\)}$ is a finite-dimensional algebra) and output = "list", then a list of vectors is returned. In this case the vector fields in the center of $\mathrm{\Γ}$are removed. With output = "pde", the determining differential equations for the normalizer are returned.

With the keyword argument ansatz = Z, the procedure InfinitesimalPseudoGroupNormalizer calculates Nor($$$\mathrm{\Γ}\, \mathrm{\𝒢}\mathit{\)}$, where$\mathit{ }\mathrm{\𝒢}$ is the infinitesimal pseudo-group defined by the vector field Z. With this keyword option, the unknown functions in Z must be explicitly listed using the keyword argument unknowns = U. Additional algebraic and differential constraints on the unknown functions in Z may be specified with the keyword argument auxiliaryequations = E. Note that the full system of differential equations for Z is likely to be inconsistent if the vector fields in $\mathrm{\Γ}$do not satisfy the differential constraints defined by E.

 If the abstract Lie algebra determined by $\mathrm{\Γ}$has been calculated and initialized (see LieAlgebraData, DGsetup), then this information can be passed to InfinitesimalPseudoGroupNormalizer with the keyword argument liealgebra = name.

 The keyword argument parameters = P will invoke the case-splitting capabilities of pdsolve . In this case, the output of InfinitesimalPseudoGroupNormalizer will be a sequence of normalizers for both the generic and special values of the parameters.

The command InfinitesimalPseudoGroupNormalizer is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form InfinitesimalPseudoGroupNormalizer(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-InfinitesimalPseudoGroupNormalizer(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{GroupActions}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$$\mathrm{with}\left(\mathrm{Library}\right)\:$

$\mathrm{DGsetup}\left(\left[x\right]\,M\right)\:$

$\mathrm{\Γ11}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\right]\right)$

$\mathrm{\Γ11}:=\left[\mathrm{D\_x}\right]$

$\mathrm{\Γ12}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,x\mathrm{D\_x}\right]\right)$

$\mathrm{\Γ12}:=\left[\mathrm{D\_x}\,x\mathrm{D\_x}\right]$

$\mathrm{N11}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{\Γ11}\right)$

$\mathrm{N11}:=\left(\mathrm{\_C1}x\+\mathrm{\_C2}\right)\mathrm{D\_x}$

$\mathrm{N11list}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{\Γ11}\,\mathrm{output}=''list''\right)$

$\mathrm{N11list}:=\left[x\mathrm{D\_x}\right]$

$\mathrm{N12}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{\Γ12}\right)$

$\mathrm{N12}:=\left[\right]$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,N\right)$

$\mathrm{frame\; name:\; N}$

$\mathrm{\Γ2}≔\mathrm{evalDG}\left(\left[z\mathrm{D\_y}-y\mathrm{D\_z}\,x\mathrm{D\_y}-y\mathrm{D\_x}\,z\mathrm{D\_x}-x\mathrm{D\_z}\right]\right)$

$\mathrm{\Γ2}:=\left[\mathrm{D\_y}z-\mathrm{D\_z}y\,-\mathrm{D\_x}y\+\mathrm{D\_y}x\,\mathrm{D\_x}z-\mathrm{D\_z}x\right]$

$\mathrm{N2}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{\Γ2}\right)$

$\mathrm{N2}:=\frac{x\mathrm{\_F1}\left(x^2\+y^2\+z^2\right)\sqrt{-z^2}\mathrm{D\_x}}{z}\+\frac{y\mathrm{\_F1}\left(x^2\+y^2\+z^2\right)\sqrt{-z^2}\mathrm{D\_y}}{z}\+\mathrm{\_F1}\left(x^2\+y^2\+z^2\right)\sqrt{-z^2}\mathrm{D\_z}$

$U≔\left[A\,B\,C\right]\left(x\,y\,z\right)$

$U:=\left[A\left(x\,y\,z\right)\,B\left(x\,y\,z\right)\,C\left(x\,y\,z\right)\right]$

$Z≔\mathrm{evalDG}\left(U\left[1\right]\mathrm{D\_x}+U\left[2\right]\mathrm{D\_y}+U\left[3\right]\mathrm{D\_z}\right)$

$Z:=A\left(x\,y\,z\right)\mathrm{D\_x}\+B\left(x\,y\,z\right)\mathrm{D\_y}\+C\left(x\,y\,z\right)\mathrm{D\_z}$

$E≔\left[\mathrm{diff}\left(A\left(x\,y\,z\right)\,x\right)+\mathrm{diff}\left(B\left(x\,y\,z\right)\,y\right)+\mathrm{diff}\left(C\left(x\,y\,z\right)\,z\right)=0\right]\:$

$\mathrm{N2div}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{\Γ2}\,\mathrm{ansatz}=Z\,\mathrm{unknowns}=U\,\mathrm{auxiliaryequations}=E\right)$

$\mathrm{N2div}:=\frac{x\mathrm{\_C1}\mathrm{D\_x}}{{\left(x^2\+y^2\+z^2\right)}^{3\/2}}\+\frac{y\mathrm{\_C1}\mathrm{D\_y}}{{\left(x^2\+y^2\+z^2\right)}^{3\/2}}\+\frac{\mathrm{\_C1}z\mathrm{D\_z}}{{\left(x^2\+y^2\+z^2\right)}^{3\/2}}$

$\mathrm{\Γ3}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\,\left(x+z\right)\mathrm{D\_x}+\mathrm{\alpha }y\mathrm{D\_y}+z\mathrm{D\_z}\right]\right)$

$\mathrm{\Γ3}:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\,\left(x\+z\right)\mathrm{D\_x}\+\mathrm{\α}y\mathrm{D\_y}\+z\mathrm{D\_z}\right]$

$\mathrm{N3}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{\Γ3}\,\mathrm{parameters}=\left\{\mathrm{\alpha }\right\}\,\mathrm{output}=''list''\right)$

$\mathrm{N3}:=\left[z\mathrm{D\_x}\,\mathrm{D\_x}x\+\mathrm{D\_z}z\,\mathrm{D\_y}\right]\,\left[z\mathrm{D\_x}\,\mathrm{D\_x}x\+\mathrm{D\_z}z\right]\,\left[\left\{\mathrm{\α}\=0\right\}\,\left\{\mathrm{\α}\=\mathrm{\α}\right\}\right]$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)$

$g:=\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}$

$\mathrm{\omega }≔\mathrm{evalDG}\left(\mathrm{dy}-z\mathrm{dx}\right)$

$\mathrm{\ω}:=-\mathrm{dx}z\+\mathrm{dy}$

$\mathrm{\Γ4}≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\,\left[\mathrm{\omega }\right]\right]\,\mathrm{output}=''list''\right)$

$\mathrm{\Γ4}:=\left[-y\mathrm{D\_x}\+x\mathrm{D\_y}\+\left(z^2\+1\right)\mathrm{D\_z}\,\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$Z≔\mathrm{evalDG}\left(\mathrm{A1}\left(x\,y\,z\right)\mathrm{D\_x}+\mathrm{A2}\left(x\,y\,z\right)\mathrm{D\_y}+\mathrm{A3}\left(x\,y\,z\right)\mathrm{D\_z}\right)$

$Z:=\mathrm{A1}\left(x\,y\,z\right)\mathrm{D\_x}\+\mathrm{A2}\left(x\,y\,z\right)\mathrm{D\_y}\+\mathrm{A3}\left(x\,y\,z\right)\mathrm{D\_z}$

$E≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\left[\mathrm{\omega }\right]\right]\,\mathrm{ansatz}=Z\,\mathrm{unknowns}=\left[\mathrm{A1}\,\mathrm{A2}\,\mathrm{A3}\right]\left(x\,y\,z\right)\,\mathrm{output}=''pde''\right)$

$E:=\left[z\mathrm{\_K111}\left(x\,y\,z\right)\+\frac{\partial }{\partial x}\mathrm{A2}\left(x\,y\,z\right)-\left(\frac{\partial }{\partial x}\mathrm{A1}\left(x\,y\,z\right)\right)z-\mathrm{A3}\left(x\,y\,z\right)\,-\mathrm{\_K111}\left(x\,y\,z\right)\+\frac{\partial }{\partial y}\mathrm{A2}\left(x\,y\,z\right)-\left(\frac{\partial }{\partial y}\mathrm{A1}\left(x\,y\,z\right)\right)z\,\frac{\partial }{\partial z}\mathrm{A2}\left(x\,y\,z\right)-\left(\frac{\partial }{\partial z}\mathrm{A1}\left(x\,y\,z\right)\right)z\,0\right]\,\left[\mathrm{A1}\left(x\,y\,z\right)\,\mathrm{A2}\left(x\,y\,z\right)\,\mathrm{A3}\left(x\,y\,z\right)\,\mathrm{\_K111}\left(x\,y\,z\right)\right]$

$\mathrm{Nor4}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{\Γ4}\,\mathrm{ansatz}=Z\,\mathrm{unknowns}=E\left[2\right]\,\mathrm{auxiliaryequations}=E\left[1\right]\,\mathrm{output}=''list''\right)$

$\mathrm{Nor4}:=\left[\mathrm{D\_x}x\+\mathrm{D\_y}y\,-\frac{z\mathrm{D\_x}}{\sqrt{z^2\+1}}\+\frac{\mathrm{D\_y}}{\sqrt{z^2\+1}}\right]$

$\mathrm{LieDerivative}\left(\mathrm{Nor4}\,\mathrm{\omega }\right)$

$\left[-\mathrm{dx}z\+\mathrm{dy}\,0\mathrm{dx}\right]$

$\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\left[\mathrm{op}\left(\mathrm{\Γ4}\right)\,\mathrm{op}\left(\mathrm{Nor4}\right)\right]\,\mathrm{alg}\right)$

$\mathrm{LD}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=-\mathrm{e5}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; alg}$

$\mathrm{Query}\left(\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\,''Ideal''\right)$

$\mathrm{true}$