Portal de Periódicos da CAPES

Intrinsic nilpotent approximation

1987; Springer Science+Business Media; Volume: 8; Issue: 3 Linguagem: Inglês

10.1007/bf00046716

1572-9036

Charles Rockland,

Nonlinear Waves and Solitons

This report is a preliminary version of work on an intrinsic approximation process arising in the context of a nonisotropic pertubation theory for certain classes of linear differential and pseudodifferential operatorsP on a manifoldM. A basic issue is that the structure ofP 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 algebrasL by (finite-dimensional) graded nilpotent Lie algebrasg x, org (x,ζ), wherex ∈ M, (x, ζ) ∈ T* M/0. It requires the notion of 'weak homomorphism'. 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 linked to 'localization', associated to an appropriate module structure onL. The analysis of the approximating operators involves the unitary representation theory of the corresponding Lie groups. These representations are for the most part infinite-dimensional, and so involve a kind of 'quantization'. Not all the representations enter. The filtered Lie algebraL leads to an 'approximate Hamiltonian action' ofG (x,ζ), the group associated tog (x,ζ), and thus induces (via an adaptation of a construction of Helffer and Nourrigat) an intrinsically defined 'asymptotic momentmap' with image ing (x,ζ) *. The relevant representations are those associated to this image by the Kirillov correspondence. The genesis of this work has been in the context of linear partial differential operators, in particular the question of hypoellipticity. For example, our framework leads to a natural hypoellipticity conjecture enlarging on that of Helffer and Nourrigat. We believe, however, that the approximation process is likely to have broader applicability, particularly in those contexts where the process can be extended to filtrations with anL 0term. This yields not simply a graded nilpotent algebra, but a semi-direct sum with a graded nilpotent. As we show, one such context arises in the approximation of nonlinear control systems.