Noncommutative geometry with graded differential Lie algebras

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 Connes-Lott prescription of noncommutative geometry, 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 x matrices. (C) 1997 American Institute of Physics.