Let $\mathrm{\Γ}$be a Lie algebra of vector fields on a manifold $M$and let$$$p \in$$M\.$$$The isotropy subalgebra  ${\mathrm{\Γ}}_p$of the Lie algebra of vector fields$\mathrm{\Γ}$ at the point $p$is defined by ${\mathrm{\Γ}}_p \= \{X \in \mathrm{\Γ} \|X_p\= 0\}$. The Lie bracket of vector fields defines a natural representation $\mathrm{\ρ}$of ${\mathrm{\Γ}}_p$ on the tangent space $T_{p}M$ by $\mathrm{\ρ}\left(X\right)\left(Y\right) \= \left[X\, \bar{Y}\right]$ for $X \in {\mathrm{\Γ}}_p$ ,$Y \in T_{p}M$ and $\bar{Y}$ any vector field on $M$ such that ${\bar{Y}}_p \=Y$. The representation $\mathrm{\ρ}$ is called the linear isotropy representation.

IsotropySubalgebra(Gamma, p) returns a list of vectors whose span defines the isotropy subalgebra ${\mathrm{\Gamma }}_p$as a subalgebra of  $\mathrm{\Γ}$.

With output = ["Vector", "Representation"], two lists are returned. The first is a list of vectors giving the isotropy subalgebra ${\mathrm{\Γ}}_p$as a subalgebra of  $\mathrm{\Γ}$and the second is the list of matrices defining the linear isotropy representation with respect to the standard basis for $T_{p}M$.

Let algname be the name of the abstract Lie algebra $\mathrm{\𝔤}\mathit{ }$created from $\mathrm{\Γ}$. With output = ["Vector", algname], the second list returned gives the isotropy subalgebra as a subalgebra of the abstract Lie algebra $\mathrm{\𝔤}$.

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

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

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

$G≔\mathrm{Retrieve}\left(''Gonzalez-Lopez''\,1\,\left[5\right]\,\mathrm{manifold}=M\right)$

$G:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_x}x-\mathrm{D\_y}y\,y\mathrm{D\_x}\,x\mathrm{D\_y}\right]$

$L≔\mathrm{LieAlgebraData}\left(G\,\mathrm{Alg1}\right)$

$L:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=-2\mathrm{e4}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=2\mathrm{e5}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=-\mathrm{e3}\right]$

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

$\mathrm{Lie\; algebra:\; Alg1}$

$\mathrm{MultiplicationTable}\left(''LieTable''\right)$

$\mathrm{Iso1}≔\mathrm{IsotropySubalgebra}\left(G\,\left[x=0\,y=0\right]\right)$

$\mathrm{Iso1}:=\left[\mathrm{D\_x}x-\mathrm{D\_y}y\,y\mathrm{D\_x}\,x\mathrm{D\_y}\right]$

$\mathrm{Iso1},\mathrm{A1}≔\mathrm{IsotropySubalgebra}\left(G\,\left[x=0\,y=0\right]\,\mathrm{output}=\left[''Vector''\,\mathrm{Alg1}\right]\right)$

$\mathrm{Iso1}\,\mathrm{A1}:=\left[\mathrm{D\_x}x-\mathrm{D\_y}y\,y\mathrm{D\_x}\,x\mathrm{D\_y}\right]\,\left[\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\right]$

$\mathrm{Iso1},\mathrm{A1},\mathrm{S1}≔\mathrm{IsotropySubalgebra}\left(G\,\left[x=0\,y=0\right]\,\mathrm{output}=\left[''Vector''\,\mathrm{Alg1}\,''Representation''\right]\right)$

$\mathrm{A1}≔\mathrm{IsotropySubalgebra}\left(G\,\left[x=0\,y=0\right]\,\mathrm{output}=\left[\mathrm{Alg1}\right]\right)$

$\mathrm{A1}:=\left[\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\right]$

$\mathrm{Iso2},\mathrm{A2},\mathrm{S2}≔\mathrm{IsotropySubalgebra}\left(G\,\left[x=1\,y=1\right]\,\mathrm{output}=\left[''Vector''\,\mathrm{Alg1}\,''Representation''\right]\right)$

$\mathrm{Query}\left(\mathrm{A1}\,\left[\mathrm{e1}\,\mathrm{e2}\right]\,''SymmetricPair''\right)$

$\mathrm{true}$

$\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{A1}\,\mathrm{iso1}\right)$

$\mathrm{L2}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-2\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=2\mathrm{e3}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=-\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\right)$

$\mathrm{Lie\; algebra:\; iso1}$

$\mathrm{\rho }≔\mathrm{Representation}\left(\mathrm{iso1}\,M\,\mathrm{S1}\right)$

$\mathrm{Query}\left(\mathrm{\rho }\,''Representation''\right)$

$\mathrm{true}$