Symbols in Noncommutative Geometry

In this paper we prove that the classical Lie bracket of vector fields can be generalized to the noncommutative setting by antisymmetrizing (in a suitable noncommutative sense) their compositions. This construction turns out to depend on the representability of linear differential operators, as it relies on the interpretation of vector fields as differential operators. In particular we provide necessary and sufficient conditions for (noncommutative) jet modules to be representing objects for differential operators. Furthermore, the primary ingredient for guaranteeing the closure of a bracket operation is a treatment of symbols, which classically represent, in an intrinsic way, the highest-order term of a differential operator. Thus, we provide an extensive theory of symbols herein.