**Connection**** on module over a graded
q-differential algebra**

Viktor Abramov

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 d^N=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.