Let mu: G x M -> M define a (right) action of an r-dimensional Lie group G on a manifold M.  

Let R_i, i = 1 ... r denote a basis for the right invariant vector fields on G.  Then the vector fields X_i(x) = mu_*(e, x)(R_i(e)) (where mu_* is the Jacobian of mu, e is the identity element of G, and x is a point of M) define a Lie algebra of vector fields on M whose structure constants coincide with the structure constants of the Lie algebra of right invariant vector fields R_i. The vector fields X_i are called the infinitesimal generators for the action mu.

For convenience, the command InfinitesimalTransformation treats the action mu as a parameterized family of transformations phi: M -> M.  The infinitesimal transformations are then computed by taking the derivatives of the components of phi with respect to the group parameters and evaluating the result at the identity.  A list of vector fields is returned, one vector field for each group parameter in par.

This command is part of the DifferentialGeometry package, and so can be used in the form InfinitesimalTransformation(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-InfinitesimalTransformation.

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

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

$\mathrm{\Φ1}≔\mathrm{Transformation}\left(M\,M\,\left[x=\cos \left(\mathrm{\theta }\right)x+\sin \left(\mathrm{\theta }\right)y+a\,y=-\sin \left(\mathrm{\theta }\right)x+\cos \left(\mathrm{\theta }\right)y+b\right]\right)$

$\mathrm{\Φ1}≔\left[x=\cos \left(\mathrm{\theta }\right)x+\sin \left(\mathrm{\theta }\right)y+a\,y=-\sin \left(\mathrm{\theta }\right)x+\cos \left(\mathrm{\theta }\right)y+b\right]$

$\mathrm{\Γ1}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{\Φ1}\,\left[a\,b\,\mathrm{\theta }\right]\right)$

$\mathrm{\Γ1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,y\mathrm{D\_x}-x\mathrm{D\_y}\right]$

$\mathrm{LieAlgebraData}\left(\mathrm{\Γ1}\right)$

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

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

$\mathrm{\Φ2}≔\mathrm{Transformation}\left(M\,R\,\left[x=\frac{ax+b}{cx+d}\right]\right)$

$\mathrm{\Φ2}≔\left[x=\frac{ax+b}{cx+d}\right]$

$\mathrm{\Γ2}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{\Φ2}\,\left[a\,b\,c\,d\right]\,\mathrm{initialpoint}=\left[a=1\,d=1\right]\right)$

$\mathrm{\Γ2}≔\left[x\mathrm{D\_x}\,\mathrm{D\_x}\,-x^2\mathrm{D\_x}\,-x\mathrm{D\_x}\right]$

$\mathrm{\Γ2}≔\mathrm{DGbasis}\left(\mathrm{\Γ2}\,\mathrm{method}=''real''\right)$

$\mathrm{\Γ2}≔\left[x\mathrm{D\_x}\,\mathrm{D\_x}\,-x^2\mathrm{D\_x}\right]$

$G≔\mathrm{solve}\left(ad-bc=1\,\left\{d\right\}\right)$

$G≔\left\{d=\frac{bc+1}{a}\right\}$

$\mathrm{\Φ3}≔\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{\Φ2}\,G\right)\right)$

$\mathrm{\Φ3}≔\left[x=\frac{\left(ax+b\right)a}{1+\left(ax+b\right)c}\right]$

$\mathrm{\Γ3}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{\Φ3}\,\left[a\,b\,c\right]\,\mathrm{initialpoint}=\left[a=1\right]\right)$

$\mathrm{\Γ3}≔\left[2x\mathrm{D\_x}\,\mathrm{D\_x}\,-x^2\mathrm{D\_x}\right]$

$\mathrm{\Φ4}≔\mathrm{Transformation}\left(M\,M\,\left[x=\frac{ax+by+c}{rx+sy+t}\,y=\frac{dx+ey+f}{rx+sy+t}\right]\right)$

$\mathrm{\Φ4}≔\left[x=\frac{ax+by+c}{rx+sy+t}\,y=\frac{dx+ey+f}{rx+sy+t}\right]$

$\mathrm{\Γ4}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{\Φ4}\,\left[a\,b\,c\,d\,e\,f\,r\,s\,t\right]\,\mathrm{initialpoint}=\left[a=1\,e=1\,t=1\right]\right)$

$\mathrm{\Γ4}≔\left[x\mathrm{D\_x}\,y\mathrm{D\_x}\,\mathrm{D\_x}\,x\mathrm{D\_y}\,y\mathrm{D\_y}\,\mathrm{D\_y}\,-x^2\mathrm{D\_x}-yx\mathrm{D\_y}\,-yx\mathrm{D\_x}-y^2\mathrm{D\_y}\,-x\mathrm{D\_x}-y\mathrm{D\_y}\right]$

$\mathrm{\Γ5}≔\mathrm{DGbasis}\left(\mathrm{\Γ4}\,\mathrm{method}=''real''\right)$

$\mathrm{\Γ5}≔\left[x\mathrm{D\_x}\,y\mathrm{D\_x}\,\mathrm{D\_x}\,x\mathrm{D\_y}\,y\mathrm{D\_y}\,\mathrm{D\_y}\,-x^2\mathrm{D\_x}-yx\mathrm{D\_y}\,-yx\mathrm{D\_x}-y^2\mathrm{D\_y}\right]$

$\mathrm{LieAlgebraData}\left(\mathrm{\Γ5}\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e2}\right]=-\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e3}\right]=-\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e4}\right]=\mathrm{e4}\,\left[\mathrm{e1}\,\mathrm{e7}\right]=\mathrm{e7}\,\left[\mathrm{e2}\,\mathrm{e4}\right]=-\mathrm{e1}+\mathrm{e5}\,\left[\mathrm{e2}\,\mathrm{e5}\right]=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e6}\right]=-\mathrm{e3}\,\left[\mathrm{e2}\,\mathrm{e7}\right]=\mathrm{e8}\,\left[\mathrm{e3}\,\mathrm{e4}\right]=\mathrm{e6}\,\left[\mathrm{e3}\,\mathrm{e7}\right]=-2\mathrm{e1}-\mathrm{e5}\,\left[\mathrm{e3}\,\mathrm{e8}\right]=-\mathrm{e2}\,\left[\mathrm{e4}\,\mathrm{e5}\right]=\mathrm{e4}\,\left[\mathrm{e4}\,\mathrm{e8}\right]=\mathrm{e7}\,\left[\mathrm{e5}\,\mathrm{e6}\right]=-\mathrm{e6}\,\left[\mathrm{e5}\,\mathrm{e8}\right]=\mathrm{e8}\,\left[\mathrm{e6}\,\mathrm{e7}\right]=-\mathrm{e4}\,\left[\mathrm{e6}\,\mathrm{e8}\right]=-\mathrm{e1}-2\mathrm{e5}\right]$