Integral Calculus on Quantum Exterior Algebras

Hom-connections and associated integral forms have been introduced and studied by T.Brzezi'nski as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus $(\mathrm{\Omega },d)$ over an algebra $A$ yields the integral complex which for various algebras has been shown to be isomorphic to the noncommutative de Rham complex (in the sense of Brzezi'nski et al.). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra $A$ with a flat hom-connection. We specialise our study to the case where an $n$-dimensional differential calculus can be constructed on a quantum exterior algebra over an $A$-bimodule. Criteria are given for free bimodules with diagonal or upper triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin’s quantum $n$-space.