L-prolongations of graded Lie algebras

TL;DR

This paper provides computational criteria for determining the finiteness of effective prolongations of fundamental graded Lie algebras, with applications to geometric structures defined by contact distributions.

Contribution

It translates Tanaka's theorem conditions into explicit matrix rank criteria, enabling practical computation of prolongation properties.

Findings

Derived explicit matrix rank conditions for prolongation finiteness.

Applicable to geometries with structure algebra on contact distributions.

Facilitates computational analysis of graded Lie algebra prolongations.

Abstract

In this paper we translate the necessary and sufficient conditions of Tanaka's theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution.