# Noncommutative $ε$-graded connections

### Axel de Goursac

Laboratoire de physique théorique d'Orsay### Thierry Masson

Laboratoire de physique theorique d'Orsay### Jean-Christophe Wallet

Laboratoire de physique theorique d'Orsay

## Abstract

We introduce the new notion of $ε$-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 $ε$-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 $ε$-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.

## Cite this article

Axel de Goursac, Thierry Masson, Jean-Christophe Wallet, Noncommutative $ε$-graded connections. J. Noncommut. Geom. 6 (2012), no. 2, pp. 343–387

DOI 10.4171/JNCG/94