Maurer–Cartan forms and the structure of Lie pseudo-groups |
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Authors: | Peter J Olver Juha Pohjanpelto |
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Institution: | (1) School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA;(2) Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA |
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Abstract: | This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional
Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan
forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the
theory of exterior differential systems and prolongation.
The second paper 60] will apply these constructions in order to develop the moving frame algorithm for the action of the
pseudo-group on submanifolds. The third paper 61] will apply Gr?bner basis methods to prove a fundamental theorem on the
freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera
for generating systems of differential invariants and also their syzygies.
Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants
and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles,
and solving equivalence and symmetry problems arising in geometry and physics. |
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Keywords: | 58A15 58A20 58H05 58J70 |
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