Séminaire du 21 septembre 2009, 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.
Virginie Collette
Last modified: Mon Mar 30 14:53:58 CEST 2009