The Structure of Continuous Pseudogroups

Monday, July 24, 2006 - 10:00am - 10:45am
EE/CS 3-180
Juha Pohjanpelto (Oregon State University)
I will report on my ongoing joint work with Peter Olver on
developing systematic and constructive algorithms for analyzing
the structure of continuous pseudogroups and identifying various
invariants for their action.

Unlike in the finite dimensional case, there is no generally
accepted abstract object to play the role of an infinite dimensional
pseudogroup. In our approach we employ the bundle of jets of
group transformations to parametrize a pseudogroup, and we realize
Maurer-Cartan forms for the pseudogroup as suitably invariant forms
on this pseudogroup jet bundle. Remarkably, the structure equations
for the Maurer-Cartan forms can then be derived from the determining
equations for the infinitesimal generators of the pseudogroup
action solely by means of linear algebra.

A moving frames for general pseudogroup actions is defined as
equivariant mappings from the space of jets of submanifolds into the
pseudogroup jet bundle. The existence of a moving frame requires local
freeness of the action in a suitable sense and, as in the finite
dimensional case, moving frames can be used to systematically produce
complete sets of differential invariants and invariant coframes for the
pseudogroup action and to effectively analyze their algebraic structure.

Our constructions are equally applicable to finite dimensional Lie group
actions and provide a slight generalization of the classical moving frame
methods in this case.