Connection on module over a graded q-differential algebra 1

Abstract

We study a concept of a q-connection on a left module, where q is a primitive Nth root of unity. This concept is based on a notion of a graded q-differential algebra whose differential d satisfies dN = 0. We propose a notion of a graded q-differential algebra with involution and making use of this notion we introduce and study a concept of a q-connection consistent with a Hermitian structure of a left module. Assuming module to be a finitely generated free module we define the components of q-connection and show that these components with respect to different basises are related by gauge transformation. We also derive the relation for components of a q-connection consistent with Hermitian structure of a module.