ec 2 00 3 A utom orphism s ofassociative algebras and noncom m utative geom etry (original) (raw)

A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the h-deformed plane and the quantum group GL p,q (2) are recovered in this way. Geometric structures like metrics and compatible linear connections are introduced. 1 Let us consider a change of basis θ s → θ ′ s := s ′ ∈S U s s ′ θ s ′ where U is an invertible matrix with entries in A, or some extension of A. Then (1.1) holds with the substitution Φ(f) → Φ ′ (f) := U Φ(f) U −1. The problem is to find a U such that Φ ′ (f) is diagonal for all f ∈ A.