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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:

(*)        VP := {P(z) d/dz  |  P(z) ∊ C[z]}   and   VR := {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 VP and/or VR 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 VP and VR ?

  • References to the above matters would also be appreciated.

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The Lie algebra you've denoted $V_P$ is the complexification of the Lie algebra of the poly vector fields on the real line, $W_1,$ which is one the 4 families of Cartan's transitive pseudogroups. While AFAIK $V_R$ doesn't have another name, its subalgebra consisting of the Laurent polynomials in $z$ times $d/dz$ is the complexification of the Lie algebra of polynomial vector fields on the circle. In a certain sense, it corresponds to the diffeomorphism group of the circle, whereas $W_1$ corresponds to the diff group of the line or the group of holomorphic automorphisms of the closed unit disk. –  Victor Protsak Aug 26 '10 at 1:25
    
Drinfeld-Krichever construction in the theory of integrable systems involves a generalization of the polynomial vector fields: for a complete algebraic curve $X,$ a point $x_\infty$ and a local coordinate $z$ at this point, consider the Lie algebra of vector fields on $X\setminus x_{\infty}$ expressed in the coordinate $z.$ Its analytic version is the Lie algebra of complex vector fields on a small circle $\partial D_\infty$ around $x_{\infty}$ that extend to holomorphic vector fields on its exterior $X\setminus D_\infty.$ –  Victor Protsak Aug 26 '10 at 1:38
    
The answer to your first question is yes. If you differentiate the action of the group $PSL_2$ on the Riemann sphere at the identity to give a Lie homomorphism from the Lie algebra $\mathfrak{sl}_2$ to vector fields on the Riemann sphere. The image of this map on a coordinate chart obtained by removing a point at infinity is your Lie algebra of complex quadratic vector fields. –  Simon Wadsley Aug 26 '10 at 16:13
    
What you mean exactly under "Lie algebras of vector fields"? If any Lie algebra admitting realization as vector fields with polynomial/rational/etc. coefficients (i.e. subalgebras of appropriate $W_n$) - there are many. This topic goes back to Sophus Lie with a tremendous amount of more recent literature. But your examples probably suggest that you are interested in "full", in some sense, algebras -- of the form $f(z) d/dz$ ? what about many variables? -- but the exact meaning is not clear to me. –  Pasha Zusmanovich Jan 5 '11 at 22:31
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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.

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