
Journal of Lie Theory 31 (2021), No. 1, 029062 Copyright Heldermann Verlag 2021 A LieTheoretic Construction of CartanMoser Chains Joel Merker Lab. de Mathématiques d'Orsay, CNRS, Université ParisSaclay, Orsay, France joel.merker@universiteparissaclay.fr [Abstractpdf] Let $M^3 \subset \mathbb{C}^2$ be a realanalytic Levi nondegenerate hypersurface. In the literature, CartanMoser chains are detected from rather advanced considerations: either from the construction of a Cartan connection associated with the CR equivalence problem; or from the construction of a formal or converging Poincar\'eMoser normal form. \par This note provides an alternative direct elementary construction, based on the inspection of the Lie prolongations of $5$ infinitesimal holomorphic automorphisms to the space of second order jets of CRtransversal curves. Within the $4$dimensional jet fiber, the orbits of these $5$ prolonged fields happen to have a simple cubic $2$dimensional degenerate exceptional orbit, the {\it chain locus:} \[ \Sigma_0\,:=\,\big\{(x_1,y_1,x_2,y_2)\in\mathbb{R}^4\colon\,\, x_2=2x_1^2y_12y_1^3,\,\,\,y_2=2x_1y_1^2+2x_1^3\big\}. \] Using plain translations, we may capture all points by working {\em only at one point}, the origin, and computations become conceptually enlightening and simple. Keywords: Lie prolongations of vector fields, CauchyRiemann manifolds, local biholomorphic equivalences, formal and convergent normal forms. MSC: 32V40, 58K50, 34C20, 14R20; 53A55, 53B25, 14B10, 5308, 53C30, 58K40, 58J70, 34C14, 58A30. [ Fulltextpdf (326 KB)] for subscribers only. 