ALGEBRAIC AND GEOMETRIC STRUCTURES IN CURRENT ALGEBRA THEORY.

The paper begins the general mathematical study of current algebras by elementary particle physicists. They are defined abstractly as infinite dimensional real Lie algebras, whose underlying vector space is a module over the ring of test-functions. Preliminary remarks about classification of these objects and general geometric and algebraic methods of their construction are made. As preparation, certain facts about differential operators on arbitrary modules are presented. In addition, remarks about the differential-geometric nature of the energy-momentum tensor and conformal symmetry are made. Finally, a global transformation group is constructed whose Lie algebra is one of the simplest current algebras.