
Journal of Lie Theory 18 (2008), No. 4, 869895 Copyright Heldermann Verlag 2008 Structure Equations of Lie PseudoGroups Francis Valiquette School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. valiq001@math.umn.edu In 1904, Elie Cartan developed a new structure theory for Lie pseudogroups based on his theory of exterior differential systems (Sur la structure des groupes infinis de transformations, in: Oeuvres Complète, Part. II, vol. 2. GauthierVillars, Paris, 1953, 571714). About a century later, in 2005, Olver and Pohjanpelto proposed a new approach to derive the structure equations of Lie pseudogroups (MaurerCartan equations and structure of Lie pseudogroups, Selecta Math. 11 (2005) 99126). The two theories are compared and it is shown that for intransitive Lie pseudogroups they do not agree. To make the two theories compatible, we show that Cartan's structure equations must be restricted to the orbits of the pseudogroup action. The repercussion of this modification on Cartan's concept of essential invariants is discussed. Also, the infinitesimal interpretation of Cartan's structure equations for transitive Lie pseudogroups, given in 1965 by Singer and Sternberg (The infinite groups of Lie and Cartan I: The transitive groups, J. d'Analyse Math. 15 (1965) 1115) is extended to intransitive Lie pseudogroups. Keywords: Lie pseudogroups, MaurerCartan structure equations, essential invariants. MSC: 58A15, 58A20, 58H05 [ Fulltextpdf (269 KB)] for subscribers only. 