In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:

            {(a z² + b z + c) d/dz  |  (a,b,c) ∊ ℂ³}

form precisely the Lie algebra whose nonzero elements generate the linear fractional transformations, i.e., PSL(2,ℂ).

• Is there some underlying reason for this? (Beyond direct calculation, which provides an easy proof.)

Other than the further Lie algebras {0}, {a d/dz}, and {(a z + b) d/dz} over ℂ of trivial, constant, and linear (affine) vector fields, there seem to be no other polynomial finite-dimensional Lie algebras of vector fields on ℂ.

• Is there some easy-to-explain reason for this?

The next "simplest" such polynomial Lie algebras of vector fields on C seem to be those defined by all polynomial and all rational functions:

(*)        V_P := {P(z) d/dz  |  P(z) ∊ C[z]}   and   V_R := {R(z) d/dz  |  R(z) ∊ C(z)}.

• Contrariwise, do there exist finite-dimensional Lie algebras of vector fields on ℂ defined by rational functions -- other than the polynomial ones mentioned above?

• In case V_P and/or V_R generate well-studied (infinite-dimensional) Lie groups of transformations of ℂ, then what are these groups?

• In any case, are there standard names for the Lie algebras V_P and V_R ?

• References to the above matters would also be appreciated.