Intrinsic nilpotent approximation

This report is a preliminary version of work on an intrinsic approximation process arising in the context of a non-isotropic perturbation theory for certain classes of linear differential and pseudodifferential operators P on a manifold M. A basic issue is that the structure of P itself determines the minimal information that the initial approximation must contain. This may vary from point to point, and requires corresponding approximate state spaces or phase spaces. This approximation process is most naturally viewed from a seemingly abstract algebraic context, namely the approximation of certain infinite-dimensional filtered Lie algebras L by (finite-dimensional) graded nilpotent Lie algebras gx, or g(x,) ' where x e M, (x,V) a T*M/O. It requires the notion of 'weak homomorphisms. A distinguishing feature of this approach is the intrinsic nature of the approximation process, in particular the minimality of the approximating Lie algebras. The process is closely linke