Shigeru Takeuchi
Lie algebras of polynomial vector fields and the invariant holomorphic functions
The purpose of this talk is to consider finite dimensional complex Lie algebras of polynomial vector fields on the complex affine space $\mathbb C^n$ and to investigate its Lie algebra structure. We would like to discuss some characteristics of the invariant holomorphic functions with respect to the action of the associated Lie groups $G$. This is an extended version of the one variable case (2010), where we have classified all the possible Lie algebras, and hence the realization as Lie groups acting effectively on the affine line. In our case the situations are more complicated if dimensions become greater. We could express $G$ invariance of a function $f$ by a system of linear partial differential equations with holomorphic coefficients, specifically algebraic ones in our case. Then we will show the possible Lie algebras. It is desirable to treat more general vector fields, i.e. with non-algebraic holomorphic coefficients. We will discuss the possibility to obtain any results in this direction.