Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups

Question

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 from open sets of ℂ into open sets, then what are these groups? Properly, these are pseudogroups, but perhaps they behave like Lie groups.

[Note: It's not hard to compute formulas for such transformations -- the flows -- directly from an expression for the vector field in terms of its zeroes (and poles, if any).]

• 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.

Answer 1

There is a local statement: Suppose you have a finite-dimensional vector space of germs of tangent vector fields at a point in a one-dimensional complex manifold, and suppose that some element of it is nonzero at that point. There is a coordinate function $z$ such that this field is $\frac{\partial}{\partial z}$. So, expressing all the elements as $f(z)\frac{\partial}{\partial z}$ for function germs $f$, you have a finite-dimensional vector space of germs of functions of $z$, having $1$ as a member and closed under the operation $f,g\mapsto fg'-gf'$. In particular the space is closed under differentiation. This forces it to consist of exponential polynomials, in fact to be the direct sum, over some set of complex numbers $\omega$ including $0$, of the space spanned by $z^je^{\omega z}$ for $0\le j\le m(\omega)$ for some integers $m(\omega)\ge 0$. For this to be closed under that bracket operation it must be either $3$-dimensional with basis $1,z,z^2$ or $1,e^{\omega z},e^{-\omega z}$ for some $\omega\ne 0$, or $2$-dimensional with basis $1,z$ or $1,e^{\omega z}$ for some $\omega\ne 0$, or $1$-dimensional with basis $1$. Under a further change of coordinates this becomes the example you mentioned or a subalgebra thereof.

It's pretty much the same over the real numbers.

Answer 2

The germs of Lie algebra actions on curves and surfaces were classified by Lie. In particular, the action of $\mathfrak{sl}(2,\mathbb{C})$ on $\mathbb{C}$ comes from the action of $\mathbb{P}SL(2,\mathbb{C})$ on $\mathbb{CP}^1$, differentiated and written out in an affine chart. Lie proved that this action does not embed into any holomorphic effective action of any larger complex Lie group. The proof is not difficult: if you add any other holomorphic vector field $f(z)\partial_z$, you need the Lie brackets in there too, so you get $z^2f'(z)\partial_z$ in there. So if $f$ vanishes at the origin to some order larger than 2, then $z^2f'(z)\partial_z$ vanishes to higher order still, so you get an infinite dimensional Lie algebra. If $f$ vanishes somewhere to order larger than $2$, use the transitivity of the Lie group action. Moreover, the same has to hold out at $z=\infty$, so you can easily prove that $f$ is a polynomial of degree 2. For more about what little I know about the generalization to complex surfaces, see my paper: Complex homogeneous surfaces