An Algebraic Remark on Clifford Algebras and Differential Operators

Abstract

In this note we shall use some techniques of commutative algebra and algebraic geometry to give a characterization of linear differential operators acting on an abstract Clifford algebra. Mathematics Subject Classification (2000). Primary 13N10; Secondary 11E88.Keywords. Linear differential operators, Clifford algebras.The ideas and results are presented here in an algebraic way. However the scenario behind is always that of Clifford bundle on a Riemannian (or semiRiemannian) manifold. In this way we avoid pathological Clifford algebras and questions about existence of objects in a more general situation.The differential operators are studied by techniques of commutative algebra and algebraic geometry, which reveals the role played by derivations. The triviality of some cohomologies for the Clifford algebra gives then a characterization of a special kind of derivations. Thus, as a consequence, we get a characterization of linear differential operators.As this paper is intended to be a note its style is close to telegraphic: formulation of definition and results plus minimal clarification. For this reason, in many places we shall not mention the necessary motivation and examples. Nevertheless the necessary references are provided.