The Mathematical Footing of Non-associative Geometry

Abstract

Starting with a Hilbert space endowed with a representation of a unitary Lie algebra and an action of a generalized Dirac operator, we develop a mathematical concept towards gauge field theories. This concept shares common features with the non--commutative geometry a la Connes/Lott, differs from that, however, by the implementation of unitary Lie algebras instead of associative *-algebras. The general scheme is presented in detail and is applied to functions $\otimes$ matrices.

39 pages, LaTeX2e + AMS macros, revised version: modified definition of an ideal, because the old definition leads to a vanishing Higgs potential in the standard model (hep-th/9607096)