Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN

Abstract

We consider the significant class of holomorphically nondegenerate CR manifolds of finite type that are represented by some weighted homogeneous polynomials and we derive some useful features which enable us to set up a fast effective algorithm to compute their Lie algebras of infinitesimal CRautomorphisms. 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 some other 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. Moreover, we consider the new and effective concept of comprehensive Gröbner systems which provides us some powerful tools to treat the computations in the parametric cases. The designed algorithm is also implemented in the MAPLE software.

Biholomorphic equivalence to totally nondegenerate model CR manifolds

“…Putting these expressions in S and multiplying again the appearing fractional equations by some sufficient powers of a 1 and a 1 , then one finds S as a weighted homogeneous polynomial system with no any parameter. Except a 2 and a 3 that we already spent their associated equations (31) to find the expressions of the parameters t 1 and t 2 , for each other involving group parameters a • there exists one equation in S that expresses it in terms of some lower weight group parameters. Our next goal is to provide two more polynomial equations including a 2 and a 3 to recover this constraint.…”

“…The collection of all infinitesimal CR automorphisms associated with M k form a Lie algebra, denoted by aut CR (M k ). It is the CR symmetry Lie algebra of M k in the terminology of Sophus Lie's symmetry theory [20] and is of finite dimension, of polynomial type and graded -in the sense of Tanaka -of the form ( [4,31]):…”

“…From a computational point of view, although computing the nonpositive part g − ⊕ g 0 of aut CR (M k ) is partly convenient -in particular by means of the algorithm designed in [31] -unfortunately for g + one needs highly complicated computations which rely on constructing and solving certain arising systems of partial differential equations ( [18,23,30,32]). Nevertheless, after several years of experience in computing these algebras in various dimensions, Beloshapka in [2] rigidity; that is: in its associated graded Lie algebra aut CR (M ), the subalgebra g + is trivial or equivalently ̺ = 0 (cf.…”

On the maximum conjecture

“…As is known (cf. [4,19]), this algebra is finite dimensional, of polynomial type and graded of the form:…”

Moduli spaces of model real submanifolds: Two alternative approaches

“…In particular, the size of the computations grows extensively as soon as the number of the CR-dimension or codimension grows even by one. Nevertheless, recently in [20] we have designed a powerful algorithm to compute the desired algebras by employing just some simple techniques of linear algebra and of the modern theory of comprehensive Gröbner systems instead of constructing and solving some PDE systems arising in the classical method.…”