Differential Invariant Algebras of Lie Pseudo–Groups

The goal of this paper is to describe, in as much detail as possible, the structure of the algebra of differential invariants of a Lie pseudo-group. Under the assumption of local freeness of the prolonged pseudo-group action, we develop algorithms for locating a finite generating set of differential invariants, establishing the recurrence relations for the differentiated invariants, and fixing a finite system of generating differential syzygies. In particular, if the pseudo-group acts transitively on the base manifold, then the algebra of differential invariants forms a rational, non-commutative differential algebra. We show that the essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules: the symbol module associated with the infinitesimal determining system of the pseudo-group, and the new “prolonged symbol module” constructed from the symbols of the prolonged pseudo-group generators. Modulo low order complications, the generating differential invariants and differential syzygies are in one-to-one correspondence with the algebraic generators and syzygies of the prolonged symbol module. Our algorithms and proofs are all constructive, and rely on combining the moving frame approach developed in earlier papers with Gröbner basis algorithms from commutative algebra. † Supported in part by NSF Grant DMS 05-05293. ‡ Supported in part by NSF Grants DMS 04–53304 and OCE 06–21134. October 29, 2007