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 endash 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 circle-times matrices. copyright 1997 American Institute of Physics