Algebra and algorithms for the classification of differential invariants
Seminar Room 1, Newton Institute
For a Lie group action known by its infinitesimals three practical descriptions of the differential algebra of differential invariants are given by generators and syzygies. The syzygies form a formally integrable system in terms of the non commuting invariant derivations. Applying a generalized differential elimination algorithm on those allows to reduce the number of generators.
The normalized and edge invariants were the focus in the reinterpretation of the moving frame method by Fels & Olver (1999) and its application to symmetry reduction by Mansfield (2001). My contribution here is first to show the completeness of a set of syzygies for the normalized invariants that can be written down with minimal information on the group action (namely the infinitesimal generators). Second, I provide the adequate general concept of edge invariants and show their generating properties. The syzygies for edge invariants are obtained by applying the algorithms for differential elimination that I generalized to non-commuting derivations. Another contibution is to exhibit the generating and rewriting properties of Maurer-Cartan invariants. Those have desirable properties from the computational point of view. They are all the more meaningful when one understands that they are the differential invariants that come into play in the moving frame method as practiced by Griffiths (1974) and differential geometers.
These contributions are easy to apply to any group action as they are implemented as the maple package AIDA, which works on top of the library DifferentialGeometry and the extension of the library diffalg to non commutative derivations.