Journal of Noncommutative Geometry


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Volume 6, Issue 2, 2012, pp. 343–387
DOI: 10.4171/JNCG/94

Published online: 2012-04-16

Noncommutative ε-graded connections

Axel de Goursac[1], Thierry Masson[2] and Jean-Christophe Wallet[3]

(1) Laboratoire de physique théorique d'Orsay
(2) Laboratoire de physique theorique d'Orsay
(3) Laboratoire de physique theorique d'Orsay

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 ([39]). 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.

Keywords: Derivation-based differential calculus, ε-graded associative algebra, ε-derivations and connections

de Goursac Axel, Masson Thierry, Wallet Jean-Christophe: Noncommutative ε-graded connections. J. Noncommut. Geom. 6 (2012), 343-387. doi: 10.4171/JNCG/94