Noncommutative ε-graded connections

Axel de Goursac (1) , Thierry Masson (2, 3) , Jean-Christophe Wallet (4)

1 UCL IRMP - Institut de Recherche en Mathématiques et Physique
2 CPT - Centre de Physique Théorique - UMR 6207
3 CPT - Centre de Physique Théorique - UMR 7332
4 LPT - Laboratoire de Physique Théorique d'Orsay [Orsay]

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 . 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 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.

Mots clés: Derivation-based differential calculus; epsilon-graded associative algebra; color algebras; epsilon-derivations; noncommutative connections.

Mots clés: noncommutative connections

Domaines: Algèbres quantiques [math.QA]; Physique mathématique [math-ph]

Axel de Goursac, Thierry Masson, Jean-Christophe Wallet. Noncommutative ε-graded connections. Journal of Noncommutative Geometry, 2012, 6 (2), pp.343-387. ⟨10.4171/JNCG/94⟩. ⟨hal-00707801⟩