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GroupActions[InfinitesimalSymmetriesOfGeometricObjectFields] - find the infinitesimal symmetries (vector fields) for a collection of vector fields, differential forms tensors, or connections

Calling Sequences

InfinitesimalSymmetriesOfGeometricObjectFields(T, option)

Parameters

T         - a list of vector fields, differential forms, tensors, connections, list of vector fields, list of differential forms, list of tensors

option    - output = "list", output = "pde", auxiliaryequations = [Delta1, Delta2,..] coefficientvariables = [x1, x2, ...], ansatz = X, unknowns = [F1, F2, ...], parameters = {a1, a2}

Description

 • Let $M$ be a manifold and let  be a list of tensor fields on $M$. Then the Lie algebra $\mathrm{Γ}$of infinitesimal symmetries of the list of tensors ${T}_{i}$ is the Lie algebra of vector fields on $M$ such that the Lie derivatives for
 • If the tensors all have the same tensorial type, say then let span. Then the Lie algebra of infinitesimal symmetries of the tensor space is Lie algebra of vector fields on $M$ such that for
 • The command InfinitesimalSymmetriesOfGeometricObjectFields(T) calculates the Lie algebra of infinitesimal symmetries of the tensors and tensor spaces in the list T. For example, if${T}_{4}$ are 4 tensor fields and T, then InfinitesimalSymmetriesOfGeometricObjectFields(T) will return the Lie algebra of vector fields such that span span $\left\{{T}_{3},{T}_{4}\right\}$.
 • The procedure InfinitesimalSymmetriesOfGeometricObjectFields creates an arbitrary vector field on and generates a system of first order PDE for the coefficients of from the Lie derivative equations and . These PDE are solved using pdsolve .
 • If the (real) Lie algebra $\mathrm{Γ}$ of infinitesimal symmetries for a given collection of geometric object fields is finite dimensional (so that the most general infinitesimal symmetry depends only upon arbitrary constants), then the optional argument output = "list" will return a basis for $\mathrm{Γ}$.
 • With the option output = "pde", just the determining differential equations for the symmetries are returned.
 • The variables appearing in the coefficients of the vector field X can be specified with the option coefficientvariables = [x1, x2, ...].
 • The exact form of the infinitesimal symmetries to be found can be specified with the option ansatz = X. With this option, the unknown coefficients to be solved for must be explicitly identified with the option unknowns = [F1, F2, ...].
 • Additional constraints on the symmetry vector field X can be specified with the optional argument auxiliaryequations = [Delta1, Delta2,..], where Delta1, Delta2,.. are differential equations whose unknowns are the coefficients of the vector field X.
 • If the given geometric object fields T depend upon parameters {a1, a2, ...}, then the optional argument parameters = {a1, a2, ...} will invoke the case splitting capabilities of pdsolve. Exceptional parameter values will be determined and a sequence of lists of infinitesimal symmetries, one list for each set of parameter values, will be returned.
 • Other optional arguments for pdsolve may be passed through the command InvariantGeometricObjectFields.
 • If pdsolve is unable to explicitly solve the pde system for the infinitesimal symmetries, then NULL is returned.
 • The command InfinitesimalSymmetriesOfGeometricObjectFields is part of the DifferentialGeometry:-GroupActions package. It can be used in the form InfinitesimalSymmetriesOfGeometricObjectFields(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-InfinitesimalSymmetriesOfGeometricObjectFields(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$

We define a manifold with coordinates.

 J > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Example 1.

Find all vector fields which commute with the vector field .

 M > $Yâ‰”\mathrm{D_x}$
 ${Y}{:=}{\mathrm{D_x}}$ (2.2)
 M > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[Y\right]\right)$
 ${\mathrm{_F3}}{}\left({y}{,}{z}\right){}{\mathrm{D_x}}{+}{\mathrm{_F2}}{}\left({y}{,}{z}\right){}{\mathrm{D_y}}{+}{\mathrm{_F1}}{}\left({y}{,}{z}\right){}{\mathrm{D_z}}$ (2.3)

Find all vector fields whose coefficients depend only on $x$ which commute with the vector field .

 M > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[Y\right],\mathrm{coefficientvariables}=\left[x\right]\right)$
 ${\mathrm{D_x}}{}{\mathrm{_C3}}{+}{\mathrm{D_y}}{}{\mathrm{_C2}}{+}{\mathrm{D_z}}{}{\mathrm{_C1}}$ (2.4)

Example 2.

Find the infinitesimal symmetries for the metric .

 M > $gâ‰”\mathrm{evalDG}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.5)
 M > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right],\mathrm{output}="list"\right)$
 $\left[{-}{\mathrm{D_y}}{}{z}{+}{\mathrm{D_z}}{}{y}{,}{\mathrm{D_z}}{,}{-}{\mathrm{D_x}}{}{z}{+}{\mathrm{D_z}}{}{x}{,}{-}{\mathrm{D_x}}{}{y}{+}{\mathrm{D_y}}{}{x}{,}{\mathrm{D_y}}{,}{\mathrm{D_x}}\right]$ (2.6)

Show the defining differential equations for these symmetries. Here we explicitly define the general form of the symmetry vector and specify the unknowns.

 M > $Xâ‰”\mathrm{evalDG}\left(R\left(x,y,z\right)\mathrm{D_x}+S\left(x,y,z\right)\mathrm{D_y}+T\left(x,y,z\right)\mathrm{D_z}\right)$
 ${X}{:=}{R}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_x}}{+}{S}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_y}}{+}{T}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_z}}$ (2.7)
 M > $Uâ‰”\left[R\left(x,y,z\right),S\left(x,y,z\right),T\left(x,y,z\right)\right]$
 ${U}{:=}\left[{R}{}\left({x}{,}{y}{,}{z}\right){,}{S}{}\left({x}{,}{y}{,}{z}\right){,}{T}{}\left({x}{,}{y}{,}{z}\right)\right]$ (2.8)
 M > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right],\mathrm{output}="pde",\mathrm{unknowns}=U,\mathrm{ansatz}=X\right)$
 $\left[{2}{}\left(\frac{{\partial }}{{\partial }{x}}{}{R}{}\left({x}{,}{y}{,}{z}\right)\right){,}\frac{{\partial }}{{\partial }{x}}{}{S}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{y}}{}{R}{}\left({x}{,}{y}{,}{z}\right){,}\frac{{\partial }}{{\partial }{x}}{}{T}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{z}}{}{R}{}\left({x}{,}{y}{,}{z}\right){,}\frac{{\partial }}{{\partial }{x}}{}{S}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{y}}{}{R}{}\left({x}{,}{y}{,}{z}\right){,}{2}{}\left(\frac{{\partial }}{{\partial }{y}}{}{S}{}\left({x}{,}{y}{,}{z}\right)\right){,}\frac{{\partial }}{{\partial }{y}}{}{T}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{z}}{}{S}{}\left({x}{,}{y}{,}{z}\right){,}\frac{{\partial }}{{\partial }{x}}{}{T}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{z}}{}{R}{}\left({x}{,}{y}{,}{z}\right){,}\frac{{\partial }}{{\partial }{y}}{}{T}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{z}}{}{S}{}\left({x}{,}{y}{,}{z}\right){,}{2}{}\left(\frac{{\partial }}{{\partial }{z}}{}{T}{}\left({x}{,}{y}{,}{z}\right)\right){,}{0}\right]{,}\left[{R}{}\left({x}{,}{y}{,}{z}\right){,}{S}{}\left({x}{,}{y}{,}{z}\right){,}{T}{}\left({x}{,}{y}{,}{z}\right)\right]$ (2.9)

We can use the auxilaryequations option to find the symmetries X of the metric g for which

 M > $\mathrm{Δ}â‰”\left[R\left(x,y,z\right)+S\left(x,y,z\right)+T\left(x,y,z\right)=0\right]$
 ${\mathrm{Δ}}{:=}\left[{R}{}\left({x}{,}{y}{,}{z}\right){+}{S}{}\left({x}{,}{y}{,}{z}\right){+}{T}{}\left({x}{,}{y}{,}{z}\right){=}{0}\right]$ (2.10)
 M > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right],\mathrm{output}="list",\mathrm{auxiliaryequations}=\mathrm{Δ},\mathrm{unknowns}=U,\mathrm{ansatz}=X\right)$
 $\left[{-}{\mathrm{D_x}}{+}{\mathrm{D_z}}{,}{-}\left({y}{-}{z}\right){}{\mathrm{D_x}}{+}\left({x}{-}{z}\right){}{\mathrm{D_y}}{-}\left({x}{-}{y}\right){}{\mathrm{D_z}}{,}{-}{\mathrm{D_x}}{+}{\mathrm{D_y}}\right]$ (2.11)

Example 3.

Find the joint infinitesimal symmetries for the 0 connection $C$ and the volume form .

 M > $Câ‰”\mathrm{Connection}\left(0&mult\left(\left(\mathrm{D_x}&tensor\mathrm{dx}\right)&tensor\mathrm{dx}\right)\right)$
 ${C}{:=}{0}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dx}}$ (2.12)
 M > $\mathrm{μ}â‰”\mathrm{evalDG}\left(\left(\mathrm{dx}&w\mathrm{dy}\right)&w\mathrm{dz}\right)$
 ${\mathrm{μ}}{:=}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (2.13)
 M > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\mathrm{μ},C\right]\right)$
 ${-}\left({\mathrm{_C2}}{}{x}{+}{\mathrm{_C8}}{}{x}{-}{\mathrm{_C9}}{}{z}{-}{y}{}{\mathrm{_C11}}{-}{\mathrm{_C10}}\right){}{\mathrm{D_x}}{+}\left({\mathrm{_C5}}{}{x}{+}{\mathrm{_C6}}{}{z}{+}{\mathrm{_C8}}{}{y}{+}{\mathrm{_C7}}\right){}{\mathrm{D_y}}{+}\left({\mathrm{_C1}}{}{x}{+}{\mathrm{_C2}}{}{z}{+}{\mathrm{_C4}}{}{y}{+}{\mathrm{_C3}}\right){}{\mathrm{D_z}}$ (2.14)

Example 4.

Here is a famous calculation due to E. Cartan. See Fulton and Harris Representation Theory page 357. We find the linear infinitesimal symmetries of the 3-form $\mathrm{ω}$ defined on the 7-manifold N with coordinates .

 M > $\mathrm{DGsetup}\left(\left[\mathrm{v1},\mathrm{v3},\mathrm{v4},\mathrm{w1},\mathrm{w3},\mathrm{w4},u\right],N\right)$
 ${\mathrm{frame name: N}}$ (2.15)
 N > $\mathrm{ω}â‰”\mathrm{evalDG}\left(\left(\mathrm{dw3}&w\mathrm{du}\right)&w\mathrm{dv3}+\left(\mathrm{dv4}&w\mathrm{du}\right)&w\mathrm{dw4}+\left(\mathrm{dw1}&w\mathrm{du}\right)&w\mathrm{dv1}+2\left(\mathrm{dv1}&w\mathrm{dv3}\right)&w\mathrm{dw4}+2\left(\mathrm{dw1}&w\mathrm{dw3}\right)&w\mathrm{dv4}\right)$
 ${\mathrm{ω}}{:=}{2}{}{\mathrm{dv1}}{}{\bigwedge }{}{\mathrm{dv3}}{}{\bigwedge }{}{\mathrm{dw4}}{+}{\mathrm{dv1}}{}{\bigwedge }{}{\mathrm{dw1}}{}{\bigwedge }{}{\mathrm{du}}{+}{\mathrm{dv3}}{}{\bigwedge }{}{\mathrm{dw3}}{}{\bigwedge }{}{\mathrm{du}}{+}{2}{}{\mathrm{dv4}}{}{\bigwedge }{}{\mathrm{dw1}}{}{\bigwedge }{}{\mathrm{dw3}}{-}{\mathrm{dv4}}{}{\bigwedge }{}{\mathrm{dw4}}{}{\bigwedge }{}{\mathrm{du}}$ (2.16)
 N > $Aâ‰”\mathrm{Matrix}\left(7,7,\left(i,j\right)→a||i||j\right)$
 N > $Xâ‰”\mathrm{convert}\left(A,\mathrm{DGvector}\right)$
 ${X}{:=}\left({\mathrm{a11}}{}{\mathrm{v1}}{+}{\mathrm{a12}}{}{\mathrm{v3}}{+}{\mathrm{a13}}{}{\mathrm{v4}}{+}{\mathrm{a14}}{}{\mathrm{w1}}{+}{\mathrm{a15}}{}{\mathrm{w3}}{+}{\mathrm{a16}}{}{\mathrm{w4}}{+}{\mathrm{a17}}{}{u}\right){}{\mathrm{D_v1}}{+}\left({\mathrm{a21}}{}{\mathrm{v1}}{+}{\mathrm{a22}}{}{\mathrm{v3}}{+}{\mathrm{a23}}{}{\mathrm{v4}}{+}{\mathrm{a24}}{}{\mathrm{w1}}{+}{\mathrm{a25}}{}{\mathrm{w3}}{+}{\mathrm{a26}}{}{\mathrm{w4}}{+}{\mathrm{a27}}{}{u}\right){}{\mathrm{D_v3}}{+}\left({\mathrm{a31}}{}{\mathrm{v1}}{+}{\mathrm{a32}}{}{\mathrm{v3}}{+}{\mathrm{a33}}{}{\mathrm{v4}}{+}{\mathrm{a34}}{}{\mathrm{w1}}{+}{\mathrm{a35}}{}{\mathrm{w3}}{+}{\mathrm{a36}}{}{\mathrm{w4}}{+}{\mathrm{a37}}{}{u}\right){}{\mathrm{D_v4}}{+}\left({\mathrm{a41}}{}{\mathrm{v1}}{+}{\mathrm{a42}}{}{\mathrm{v3}}{+}{\mathrm{a43}}{}{\mathrm{v4}}{+}{\mathrm{a44}}{}{\mathrm{w1}}{+}{\mathrm{a45}}{}{\mathrm{w3}}{+}{\mathrm{a46}}{}{\mathrm{w4}}{+}{\mathrm{a47}}{}{u}\right){}{\mathrm{D_w1}}{+}\left({\mathrm{a51}}{}{\mathrm{v1}}{+}{\mathrm{a52}}{}{\mathrm{v3}}{+}{\mathrm{a53}}{}{\mathrm{v4}}{+}{\mathrm{a54}}{}{\mathrm{w1}}{+}{\mathrm{a55}}{}{\mathrm{w3}}{+}{\mathrm{a56}}{}{\mathrm{w4}}{+}{\mathrm{a57}}{}{u}\right){}{\mathrm{D_w3}}{+}\left({\mathrm{a61}}{}{\mathrm{v1}}{+}{\mathrm{a62}}{}{\mathrm{v3}}{+}{\mathrm{a63}}{}{\mathrm{v4}}{+}{\mathrm{a64}}{}{\mathrm{w1}}{+}{\mathrm{a65}}{}{\mathrm{w3}}{+}{\mathrm{a66}}{}{\mathrm{w4}}{+}{\mathrm{a67}}{}{u}\right){}{\mathrm{D_w4}}{+}\left({\mathrm{a71}}{}{\mathrm{v1}}{+}{\mathrm{a72}}{}{\mathrm{v3}}{+}{\mathrm{a73}}{}{\mathrm{v4}}{+}{\mathrm{a74}}{}{\mathrm{w1}}{+}{\mathrm{a75}}{}{\mathrm{w3}}{+}{\mathrm{a76}}{}{\mathrm{w4}}{+}{\mathrm{a77}}{}{u}\right){}{\mathrm{D_u}}$ (2.17)
 N > $\mathrm{vars}â‰”\mathrm{convert}\left(A,\mathrm{set}\right)$
 ${\mathrm{vars}}{:=}\left\{{\mathrm{a11}}{,}{\mathrm{a12}}{,}{\mathrm{a13}}{,}{\mathrm{a14}}{,}{\mathrm{a15}}{,}{\mathrm{a16}}{,}{\mathrm{a17}}{,}{\mathrm{a21}}{,}{\mathrm{a22}}{,}{\mathrm{a23}}{,}{\mathrm{a24}}{,}{\mathrm{a25}}{,}{\mathrm{a26}}{,}{\mathrm{a27}}{,}{\mathrm{a31}}{,}{\mathrm{a32}}{,}{\mathrm{a33}}{,}{\mathrm{a34}}{,}{\mathrm{a35}}{,}{\mathrm{a36}}{,}{\mathrm{a37}}{,}{\mathrm{a41}}{,}{\mathrm{a42}}{,}{\mathrm{a43}}{,}{\mathrm{a44}}{,}{\mathrm{a45}}{,}{\mathrm{a46}}{,}{\mathrm{a47}}{,}{\mathrm{a51}}{,}{\mathrm{a52}}{,}{\mathrm{a53}}{,}{\mathrm{a54}}{,}{\mathrm{a55}}{,}{\mathrm{a56}}{,}{\mathrm{a57}}{,}{\mathrm{a61}}{,}{\mathrm{a62}}{,}{\mathrm{a63}}{,}{\mathrm{a64}}{,}{\mathrm{a65}}{,}{\mathrm{a66}}{,}{\mathrm{a67}}{,}{\mathrm{a71}}{,}{\mathrm{a72}}{,}{\mathrm{a73}}{,}{\mathrm{a74}}{,}{\mathrm{a75}}{,}{\mathrm{a76}}{,}{\mathrm{a77}}\right\}$ (2.18)
 N > $Yâ‰”\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\mathrm{ω},\mathrm{ansatz}=X,\mathrm{unknowns}=\mathrm{vars}\right)$
 ${Y}{:=}\left({\mathrm{_C1}}{}{\mathrm{v1}}{+}{\mathrm{_C2}}{}{\mathrm{v3}}{+}{\mathrm{_C3}}{}{\mathrm{v4}}{+}{\mathrm{_C4}}{}{\mathrm{w3}}{+}{\mathrm{_C5}}{}{\mathrm{w4}}{+}{\mathrm{_C6}}{}{u}\right){}{\mathrm{D_v1}}{-}\left({\mathrm{_C4}}{}{\mathrm{w1}}{-}{\mathrm{_C7}}{}{\mathrm{v1}}{-}{\mathrm{_C8}}{}{\mathrm{v3}}{-}{\mathrm{_C9}}{}{\mathrm{v4}}{-}{u}{}{\mathrm{_C11}}{-}{\mathrm{w4}}{}{\mathrm{_C10}}\right){}{\mathrm{D_v3}}{+}\left({\mathrm{_C1}}{}{\mathrm{v4}}{+}{\mathrm{_C4}}{}{u}{-}{\mathrm{_C5}}{}{\mathrm{w1}}{+}{\mathrm{_C6}}{}{\mathrm{v3}}{+}{\mathrm{_C8}}{}{\mathrm{v4}}{-}{\mathrm{v1}}{}{\mathrm{_C11}}{-}{\mathrm{w3}}{}{\mathrm{_C10}}\right){}{\mathrm{D_v4}}{-}\left({\mathrm{_C1}}{}{\mathrm{w1}}{+}{\mathrm{_C7}}{}{\mathrm{w3}}{-}{\mathrm{_C9}}{}{u}{-}{\mathrm{v3}}{}{\mathrm{_C12}}{-}{\mathrm{v4}}{}{\mathrm{_C13}}{-}{\mathrm{w4}}{}{\mathrm{_C11}}\right){}{\mathrm{D_w1}}{-}\left({\mathrm{_C2}}{}{\mathrm{w1}}{+}{\mathrm{_C3}}{}{u}{+}{\mathrm{_C6}}{}{\mathrm{w4}}{+}{\mathrm{_C8}}{}{\mathrm{w3}}{+}{\mathrm{v1}}{}{\mathrm{_C12}}{-}{\mathrm{v4}}{}{\mathrm{_C14}}\right){}{\mathrm{D_w3}}{-}\left({\mathrm{_C1}}{}{\mathrm{w4}}{+}{\mathrm{_C3}}{}{\mathrm{w1}}{+}{\mathrm{_C8}}{}{\mathrm{w4}}{+}{\mathrm{_C9}}{}{\mathrm{w3}}{+}{u}{}{\mathrm{_C12}}{+}{\mathrm{v1}}{}{\mathrm{_C13}}{+}{\mathrm{v3}}{}{\mathrm{_C14}}\right){}{\mathrm{D_w4}}{-}\left({2}{}{\mathrm{_C3}}{}{\mathrm{v3}}{-}{2}{}{\mathrm{_C4}}{}{\mathrm{w4}}{-}{2}{}{\mathrm{_C6}}{}{\mathrm{w1}}{-}{2}{}{\mathrm{_C9}}{}{\mathrm{v1}}{+}{2}{}{\mathrm{v4}}{}{\mathrm{_C12}}{-}{2}{}{\mathrm{w3}}{}{\mathrm{_C11}}\right){}{\mathrm{D_u}}$ (2.19)
 N > $câ‰”\mathrm{Tools}:-\mathrm{DGinfo}\left(Y,"NonJetIndets"\right)$
 ${c}{:=}\left\{{\mathrm{_C1}}{,}{\mathrm{_C2}}{,}{\mathrm{_C3}}{,}{\mathrm{_C4}}{,}{\mathrm{_C5}}{,}{\mathrm{_C6}}{,}{\mathrm{_C7}}{,}{\mathrm{_C8}}{,}{\mathrm{_C9}}{,}{\mathrm{_C10}}{,}{\mathrm{_C11}}{,}{\mathrm{_C12}}{,}{\mathrm{_C13}}{,}{\mathrm{_C14}}\right\}$ (2.20)
 N > $\mathrm{Gamma}â‰”\left[\mathrm{seq}\left(\mathrm{Tools}:-\mathrm{DGmap}\left(1,\mathrm{diff},Y,v\right),v=c\right)\right]$
 ${\mathrm{Γ}}{:=}\left[{\mathrm{v1}}{}{\mathrm{D_v1}}{+}{\mathrm{v4}}{}{\mathrm{D_v4}}{-}{\mathrm{w1}}{}{\mathrm{D_w1}}{-}{\mathrm{w4}}{}{\mathrm{D_w4}}{,}{\mathrm{v3}}{}{\mathrm{D_v1}}{-}{\mathrm{w1}}{}{\mathrm{D_w3}}{,}{-}{2}{}{\mathrm{D_u}}{}{\mathrm{v3}}{-}{u}{}{\mathrm{D_w3}}{+}{\mathrm{v4}}{}{\mathrm{D_v1}}{-}{\mathrm{w1}}{}{\mathrm{D_w4}}{,}{2}{}{\mathrm{D_u}}{}{\mathrm{w4}}{+}{u}{}{\mathrm{D_v4}}{-}{\mathrm{w1}}{}{\mathrm{D_v3}}{+}{\mathrm{w3}}{}{\mathrm{D_v1}}{,}{-}{\mathrm{w1}}{}{\mathrm{D_v4}}{+}{\mathrm{w4}}{}{\mathrm{D_v1}}{,}{2}{}{\mathrm{D_u}}{}{\mathrm{w1}}{+}{u}{}{\mathrm{D_v1}}{+}{\mathrm{v3}}{}{\mathrm{D_v4}}{-}{\mathrm{w4}}{}{\mathrm{D_w3}}{,}{\mathrm{v1}}{}{\mathrm{D_v3}}{-}{\mathrm{w3}}{}{\mathrm{D_w1}}{,}{\mathrm{v3}}{}{\mathrm{D_v3}}{+}{\mathrm{v4}}{}{\mathrm{D_v4}}{-}{\mathrm{w3}}{}{\mathrm{D_w3}}{-}{\mathrm{w4}}{}{\mathrm{D_w4}}{,}{2}{}{\mathrm{D_u}}{}{\mathrm{v1}}{+}{u}{}{\mathrm{D_w1}}{+}{\mathrm{v4}}{}{\mathrm{D_v3}}{-}{\mathrm{w3}}{}{\mathrm{D_w4}}{,}{-}{\mathrm{w3}}{}{\mathrm{D_v4}}{+}{\mathrm{w4}}{}{\mathrm{D_v3}}{,}{2}{}{\mathrm{D_u}}{}{\mathrm{w3}}{+}{u}{}{\mathrm{D_v3}}{-}{\mathrm{v1}}{}{\mathrm{D_v4}}{+}{\mathrm{w4}}{}{\mathrm{D_w1}}{,}{-}{2}{}{\mathrm{D_u}}{}{\mathrm{v4}}{-}{u}{}{\mathrm{D_w4}}{-}{\mathrm{v1}}{}{\mathrm{D_w3}}{+}{\mathrm{v3}}{}{\mathrm{D_w1}}{,}{-}{\mathrm{v1}}{}{\mathrm{D_w4}}{+}{\mathrm{v4}}{}{\mathrm{D_w1}}{,}{-}{\mathrm{v3}}{}{\mathrm{D_w4}}{+}{\mathrm{v4}}{}{\mathrm{D_w3}}\right]$ (2.21)
 N > $\mathrm{nops}\left(\mathrm{Gamma}\right)$
 ${14}$ (2.22)

It is a simple matter to use the package LieAlgebras to check that this Lie algebra is indecomposable and simple and is a realization of the exceptional Lie algebra ${g}_{2}$.

Example 5.

Find the point symmetries of the Lagrangian for the (2 +1) wave equation. The result is a 8-dimensional Lie algebra.

 N > $\mathrm{DGsetup}\left(\left[x,y,t\right],\left[u\right],J,1\right)$
 ${\mathrm{frame name: J}}$ (2.23)
 J > $\mathrm{λ}â‰”\mathrm{evalDG}\left(\left({u}_{1}^{2}+{u}_{2}^{2}-{u}_{3}^{2}\right)\left(\mathrm{Dx}&w\mathrm{Dy}\right)&w\mathrm{Dt}\right)$
 ${\mathrm{λ}}{:=}\left({{u}}_{{1}}^{{2}}{+}{{u}}_{{2}}^{{2}}{-}{{u}}_{{3}}^{{2}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{\mathrm{Dt}}$ (2.24)
 J > $\mathrm{Gamma}â‰”\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\mathrm{λ}\right],\mathrm{output}="list"\right)$
 ${\mathrm{Γ}}{:=}\left[{-}{2}{}{\mathrm{D_t}}{}{t}{-}{2}{}{\mathrm{D_x}}{}{x}{-}{2}{}{\mathrm{D_y}}{}{y}{+}{{\mathrm{D_u}}}_{{[}{]}}{}{{u}}_{{[}{]}}{,}{{\mathrm{D_u}}}_{{[}{]}}{,}{\mathrm{D_t}}{}{y}{+}{\mathrm{D_y}}{}{t}{,}{\mathrm{D_t}}{,}{\mathrm{D_t}}{}{x}{+}{\mathrm{D_x}}{}{t}{,}{-}{\mathrm{D_x}}{}{y}{+}{\mathrm{D_y}}{}{x}{,}{\mathrm{D_y}}{,}{\mathrm{D_x}}\right]$ (2.25)
 J > $\mathrm{nops}\left(\mathrm{Gamma}\right)$
 ${8}$ (2.26)

Example 6.

Find the infinitesimal conformal symmetries of the metric .  These are the vector fields $X$ such that  or span$\left\{g\right\}$.

 J > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.27)
 M > $gâ‰”\mathrm{evalDG}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.28)

Note that the first argument is now a list of a list.

 M > $\mathrm{ConSym}â‰”\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\left[g\right]\right],\mathrm{output}="list"\right)$
 ${\mathrm{ConSym}}{:=}\left[\left({-}\frac{{1}}{{4}}{}{{y}}^{{2}}{-}\frac{{1}}{{4}}{}{{z}}^{{2}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}\right){}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{x}{}{y}{}{\mathrm{D_y}}{+}\frac{{1}}{{2}}{}{x}{}{z}{}{\mathrm{D_z}}{,}\frac{{1}}{{2}}{}{x}{}{z}{}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{y}{}{z}{}{\mathrm{D_y}}{-}\left({-}\frac{{1}}{{4}}{}{{z}}^{{2}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{y}}^{{2}}\right){}{\mathrm{D_z}}{,}\frac{{1}}{{2}}{}{x}{}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{y}{}{\mathrm{D_y}}{+}\frac{{1}}{{2}}{}{z}{}{\mathrm{D_z}}{,}\frac{{1}}{{2}}{}{x}{}{y}{}{\mathrm{D_x}}{-}\left(\frac{{1}}{{4}}{}{{z}}^{{2}}{-}\frac{{1}}{{4}}{}{{y}}^{{2}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}\right){}{\mathrm{D_y}}{+}\frac{{1}}{{2}}{}{y}{}{z}{}{\mathrm{D_z}}{,}{-}{\mathrm{D_y}}{}{z}{+}{\mathrm{D_z}}{}{y}{,}{\mathrm{D_z}}{,}{-}{\mathrm{D_x}}{}{z}{+}{\mathrm{D_z}}{}{x}{,}{-}{\mathrm{D_x}}{}{y}{+}{\mathrm{D_y}}{}{x}{,}{\mathrm{D_y}}{,}{\mathrm{D_x}}\right]$ (2.29)

The conformal symmetries of $\mathrm{𝔤}$ define a 10-dimensional Lie algebra.

 M > $\mathrm{nops}\left(\mathrm{ConSym}\right)$
 ${10}$ (2.30)

Example 7.

Find the  infinitesimal symmetries of a distribution of vector fields $\mathrm{Δ}$. These are the vector fields such that ${\mathrm{ℒ}}_{X}$(Y)  for each .

 M > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5}\right],Q\right)$
 ${\mathrm{frame name: Q}}$ (2.31)
 Q > $\mathrm{Δ}â‰”\mathrm{evalDG}\left(\left[\mathrm{D_x1}+\mathrm{x3}\mathrm{D_x2}+\mathrm{x4}\mathrm{D_x3}+{\mathrm{x4}}^{3}\mathrm{D_x5},\mathrm{D_x4}\right]\right):$
 Q > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\mathrm{Δ}\right],\mathrm{output}="list"\right)$
 $\left[{-}{\mathrm{x2}}{}{\mathrm{D_x2}}{-}{\mathrm{x3}}{}{\mathrm{D_x3}}{-}{\mathrm{x4}}{}{\mathrm{D_x4}}{-}{3}{}{\mathrm{x5}}{}{\mathrm{D_x5}}{,}{-}\frac{{1}}{{2}}{}{{\mathrm{x4}}}^{{2}}{}{\mathrm{D_x1}}{-}\left({-}\frac{{1}}{{6}}{}{\mathrm{x5}}{+}\frac{{1}}{{2}}{}{\mathrm{x3}}{}{{\mathrm{x4}}}^{{2}}\right){}{\mathrm{D_x2}}{-}\frac{{1}}{{3}}{}{{\mathrm{x4}}}^{{3}}{}{\mathrm{D_x3}}{-}\frac{{1}}{{5}}{}{{\mathrm{x4}}}^{{5}}{}{\mathrm{D_x5}}{,}{-}{\mathrm{x1}}{}{\mathrm{D_x1}}{-}{2}{}{\mathrm{x2}}{}{\mathrm{D_x2}}{-}{\mathrm{x3}}{}{\mathrm{D_x3}}{-}{\mathrm{x5}}{}{\mathrm{D_x5}}{,}{\mathrm{D_x5}}{,}{\mathrm{x1}}{}{\mathrm{D_x2}}{+}{\mathrm{D_x3}}{,}{\mathrm{D_x2}}{,}{\mathrm{D_x1}}\right]$ (2.32)

Example 8.

Find the symmetries of a metric which depend upon 2 parameters , where .

 Q > $gâ‰”\mathrm{evalDG}\left(\mathrm{dx}&t\mathrm{dx}+{ⅇ}^{\mathrm{α}x}\mathrm{dy}&t\mathrm{dy}+\left(\mathrm{β}y+1\right)\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{ⅇ}}^{{\mathrm{α}}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}\left({y}{}{\mathrm{β}}{+}{1}\right){}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.33)
 M > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right],\mathrm{output}="list",\mathrm{parameters}=\left\{\mathrm{α},\mathrm{β}\right\},\mathrm{auxiliaryequations}=\left\{\mathrm{α}\ne 0\right\}\right)$
 $\left[{\mathrm{D_z}}{,}{-}\frac{{2}{}{\mathrm{D_x}}}{{\mathrm{α}}}{+}{y}{}{\mathrm{D_y}}{,}{\mathrm{D_y}}{,}{y}{}{\mathrm{α}}{}{\mathrm{D_x}}{-}\left(\frac{{1}}{{4}}{}{{y}}^{{2}}{}{{\mathrm{α}}}^{{2}}{-}{{ⅇ}}^{{-}{\mathrm{α}}{}{x}}\right){}{\mathrm{D_y}}\right]{,}\left[\frac{{4}{}{\mathrm{D_x}}}{{\mathrm{α}}}{-}\frac{{2}{}\left({y}{}{\mathrm{β}}{+}{1}\right){}{\mathrm{D_y}}}{{\mathrm{β}}}{+}{z}{}{\mathrm{D_z}}{,}{\mathrm{D_z}}\right]{,}\left[\left\{{\mathrm{α}}{=}{\mathrm{α}}{,}{\mathrm{β}}{=}{0}\right\}{,}\left\{{\mathrm{α}}{=}{\mathrm{α}}{,}{\mathrm{β}}{=}{\mathrm{β}}\right\}\right]$ (2.34)

Example 9.

The command InfinitesimalSymmetriesOfGeometricObjectFields can also be used to calculate the symmetries of a tensor defined on a Lie algebra.

 > $\mathrm{LD}â‰”\mathrm{Library}:-\mathrm{Retrieve}\left("Winternitz",1,\left[4,10\right],\mathrm{alg1}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.35)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: alg1}}$ (2.36)
 alg1 > $Tâ‰”\mathrm{evalDG}\left(\left(\mathrm{e4}&t\mathrm{θ1}\right)&t\mathrm{e4}\right)$
 ${T}{:=}{\mathrm{e4}}{}{\mathrm{θ1}}{}{\mathrm{e4}}$ (2.37)
 alg1 > $\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[T\right]\right)$
 ${\mathrm{_C1}}{}{\mathrm{e1}}{+}{\mathrm{_C2}}{}{\mathrm{e4}}$ (2.38)