Noncommutative epsilon-graded connections

TL;DR

This paper introduces epsilon-graded associative algebras and develops a differential calculus and connection theory for them, with applications to matrix algebras and noncommutative gauge theories.

Contribution

It proposes a new framework of epsilon-graded algebras and extends differential calculus and connections to this setting, linking to noncommutative gauge theories.

Findings

Defined epsilon-graded associative algebras and epsilon-derivations

Developed epsilon-graded differential calculus and connections

Applied to examples including matrix and Moyal algebras

Abstract

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