Integral Calculus on Quantum Exterior Algebras

TL;DR

This paper investigates the conditions under which the integral complex and the noncommutative de Rham complex are isomorphic in the context of quantum exterior algebras with flat hom-connections, providing criteria and examples.

Contribution

It introduces criteria for the isomorphism of integral and de Rham complexes on quantum exterior algebras with flat hom-connections, extending previous noncommutative geometric frameworks.

Findings

Criteria for isomorphism on free bimodules with diagonal structure

Application to quantum polynomial algebra calculus

Application to Manin's quantum n-space calculus

Abstract

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 $(\Omega, d)$ 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.