Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems

In this paper we give a method which uses a nite number of di erentiations and linear operations to determine the Cartan structure of structurally transitive Lie pseudogroups from their in nitesimal de ning equations. These equations are the linearized form of the pseudogroup de ning system { the system of pdes whose solutions are the transformations belonging to the pseudogroup. In many applications the explicit form of the transformations is not available and we only have access to their pseudogroup and in nitesimal de ning systems, neither of which is initially in involutive form. The usual method for calculating Cartan structure takes as its starting point the pseudogroup de ning system in involutive form. However there is currently no constructive algorithm which can always reduce nonlinear pseudogroup de ning systems to involutive form. In contrast there are many algorithms which can constructively reduce linear in nitesimal de ning systems to involutive form. Our method for calculating structure exploits these algorithms and the work of Singer and Sternberg on the structure of transitive Lie pseudogroups. We also give a constructive method for determining whether a Lie pseudogroup is structurally transitive from its in nitesimal de ning system. Our method makes feasible the calculation of the Cartan structure of in nite Lie pseudogroups of symmetries of di erential equations. Examples including the KP equation and Liouville's equation are given.