Covariant Differential Calculus on Quantum Spaces

Abstract

Nowadays differential forms on manifolds have entered the formulation of a number of physical theories such as Maxwell’s theory, mechanics, the theory of relativity and others. There are various physical ideas and considerations (quantum gravity, discrete space-time structures, models of elementary particle physics) that strongly motivate the replacement of the commutative algebra of C^∞-functions on a manifold by an appropriate noncommutative algebra and the study of “noncommutative geometry” there. Differential forms also appear to be a proper framework for doing this. The basic concept in this context is that of a “differential calculus” of an algebra. It allows us to introduce differential geometric notions and carries in this sense the geometry of the “noncommutative space” which may be thought to be behind the algebra.