Journal of Lie Theory 18 (2008), No. 4, 869--895
Copyright Heldermann Verlag 2008
Structure Equations of Lie Pseudo-Groups
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 pseudo-groups based on his theory of exterior differential systems (Sur la structure des groupes infinis de transformations, in: Oeuvres Complète, Part. II, vol. 2. Gauthier-Villars, Paris, 1953, 571--714). About a century later, in 2005, Olver and Pohjanpelto proposed a new approach to derive the structure equations of Lie pseudo-groups (Maurer-Cartan equations and structure of Lie pseudo-groups, Selecta Math. 11 (2005) 99--126). The two theories are compared and it is shown that for intransitive Lie pseudo-groups 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 pseudo-group 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 pseudo-groups, given in 1965 by Singer and Sternberg (The infinite groups of Lie and Cartan I: The transitive groups, J. d'Analyse Math. 15 (1965) 1--115) is extended to intransitive Lie pseudo-groups.
Keywords: Lie pseudo-groups, Maurer-Cartan structure equations, essential invariants.
MSC: 58A15, 58A20, 58H05
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