Noncommutative $\epsilon$-graded connections

Abstract

We introduce the new notion of $\epsilon$-graded associative algebras which takes its roots from the notion of commutation factors introduced in the context of Lie algebras ([Scheunert, 1979]). We define and study the associated notion of $\epsilon$-derivation-based differential calculus, which generalizes the derivation-based calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of $\epsilon$-graded commutative algebras, in particular some graded matrix algebras and the Moyal algebra. This last example also permits us to interpret mathematically a noncommutative gauge field theory.