An ODE’s Version of the Formula of Baker, Campbell, Dynkin and Hausdorff and the Construction of Lie Groups with Prescribed Lie Algebra

摘要

If is a Hörmander partial differential operator in , we give sufficient conditions on the vector fields X _j ’s for the existence of a Lie group structure (and we exhibit its construction), not necessarily nilpotent nor homogeneous, such that is left invariant on . The main tool is a formula of Baker-Campbell-Dynkin-Hausdorff type for the ODE’s naturally related to the system of vector fields {X ₀, . . . , X _m }. We provide a direct proof of this formula in the ODE’s context (which seems to be missing in literature), without invoking any result of Lie group theory, nor the abstract algebraic machinery usually involved in formulas of Baker-Campbell-Dynkin-Hausdorff type. Examples of operators to which our results apply are also furnished.