[Paper Review] Lie algebras of infinitesimal CR-automorphisms of finite type, holomorphically nondegenerate, weighted homogeneous CR-generic submanifolds of C^N

Masoud Sabzevari, Amir Hashemi+2 more|arXiv (Cornell University)|Apr 10, 2013

Holomorphic and Operator Theory31 references3 citations

TL;DR

This paper presents a fast, algorithmic method to compute the Lie algebras of infinitesimal CR-automorphisms for finite type, holomorphically nondegenerate, weighted homogeneous CR-generic submanifolds in ℂ^N. By exploiting a natural grading of the Lie algebra and using comprehensive Gröbner systems for parametric cases, the method reduces the problem to linear algebra and enables efficient computation via a divide-and-conquer strategy, implemented in Maple and verified on Beloshapka's rigid models up to length 6.

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 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 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.

Motivation & Objective

• To develop an efficient algorithm for computing the Lie algebra of infinitesimal CR-automorphisms of finite type, holomorphically nondegenerate, weighted homogeneous CR-generic submanifolds in ℂ^N.
• To overcome the computational inefficiency of traditional PDE-based methods by replacing them with linear algebra techniques.
• To handle parametric cases effectively using comprehensive Gröbner systems.
• To implement the algorithm in Maple and verify its performance on known rigid models, including Beloshapka’s conjecture up to length 6.

Proposed method

• The method exploits a natural grading of the sought Lie algebra, decomposing the computation into manageable graded components.
• It uses a divide-and-conquer strategy to break down the system into simpler sub-computations across each graded component.
• The algorithm relies on plain linear algebra techniques instead of solving complex PDE systems, significantly improving efficiency.
• Comprehensive Gröbner systems are employed to treat parametric defining equations, enabling robust computation in families of models.
• The algorithm is implemented in Maple, allowing automated computation and verification of results.
• The method is validated on Beloshapka’s rigid models, with timings reported for models of increasing complexity.

Experimental results

Research questions

• RQ1 Can the Lie algebra of infinitesimal CR-automorphisms be computed efficiently for weighted homogeneous, finite type, holomorphically nondegenerate CR submanifolds in ℂ^N?
• RQ2 How can parametric families of defining equations be handled effectively in the computation of such Lie algebras?
• RQ3 What is the computational complexity of computing these algebras as the number of variables and defining equations increases?
• RQ4 Can the algorithm verify Beloshapka’s conjecture on rigidity for models of higher length (e.g., length 6) efficiently?
• RQ5 To what extent does the use of comprehensive Gröbner systems improve the robustness and generality of the computation in parametric cases?

Key findings

• The algorithm computes the Lie algebra of infinitesimal CR-automorphisms using only linear algebra, avoiding time-consuming PDE solving.
• For Beloshapka’s models of length 5, the algorithm verified the conjecture in just 13 minutes using Maple.
• The computation time increases significantly with model complexity: 83.5 seconds for model M₁₃, rising to 1157 seconds for M₁₇ (length 6).
• The method successfully computes the full Lie algebra for 12 rigid models, including those with parametric coefficients.
• In the non-parametric case (e.g., M₁₂), the algorithm computes a 19-dimensional Lie algebra with two basis elements in 𝔤₀, confirming rigidity.
• The results confirm the rigidity of all 12 models studied, consistent with Beloshapka’s conjecture.