14h: Évelyne Hubert, GALAAD, INRIA, Sophia Antipolis - Méditerranée.

Algebra of Differential Invariants.

Whether algebraic or differential, one can distinguish two families of applications for invariants of group actions: equivalence problems, together with classification and canonical forms, and symmetry reduction. In this latter case invariants are used to take into account the symmetry of a problem, mainly in order to reduce its size or its analysis. The computational requirements include then four main components: the explicit computation of a generating set of invariants, and the relations among them (syzygies); procedures for rewriting the problem in terms of the invariants; and finally procedures for computing in the algebra of invariants. For differential invariants, those issues have been coherently addressed in a series of papers. The algebraic foundations developed therein support an algorithmic suite that is being implemented in the Maple package AIDA that works on top of the library DifferentialGeometry and the extension of the library diffalg to non commutative derivations. In this talk I focus on three descriptions of the differential algebra of differential invariants as given by generators and syzygies. The normalized and edge invariants were the focus in the reinterpretation of the moving frame method by Fels & Olver (1999). 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. The syzygies for the Maurer-Cartan invariants naturally follow from the structure equations for the group.