Symbols in Noncommutative Geometry
Keegan J. Flood, Mauro Mantegazza, Henrik Winther
TL;DR
The paper develops a robust noncommutative differential-geometry framework that generalizes the Lie bracket via antisymmetrized compositions of vector-field-like differential operators. It introduces elemental and primitive jet theories, provides representability criteria for differential-operator functors via jet objects, and builds a comprehensive symbol calculus that supports intrinsic brackets on vector fields. By integrating elemental, primitive, and Clifford-algebraic constructions, the authors establish when vector-field compositions close under a bracket and how this recovers the classical Lie bracket in the commutative limit. The work advances a full noncommutative theory of differential equations, including formal integrability notions, and supplies concrete canonical models (e.g. Clifford algebras) to illustrate the brackets and their algebraic structure.
Abstract
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.