Lie Algebras of Infinitesimal CR-Automorphisms of Weighted Homogeneous and Homogeneous CR-Generic Submanifolds of C^N

Abstract

We consider the significant class of homogeneous CR-manifolds represented by some weighted homogeneous polynomials and we derive some plain and useful features which enable us
to set up a fast effective algorithm to compute homogeneous components of their Lie algebras of infinitesimal CR-automorphisms. This algorithm mainly relies upon a natural
gradation of the sought Lie algebras, and it also
consists in treating separately the related graded components. While classical methods are based on constructing and solving an associated pde systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra.
Furthermore, it benefits from a divide-and-conquer strategy to
break down the computations into some simpler sub-computations. Also, we consider the new and effective concept of comprehensive Groebner systems which provides us
some powerful tools to treat the computations in the too much complicated parametric cases. The designed algorithm is also implemented in the Maple software. This implementation requires also implementing a recently designed algorithm of Kapur et al.