Noncommutative epsilon-graded connections

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.

37 pages

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• Noncommutative gauge theories on ${\mathbb{R}}_{\lambda }^3$: Perturbatively finite models
• Connes distance by examples: Homothetic spectral metric spaces
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• Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces
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• Fréchet Quantum Supergroups
• Spectral theorem in noncommutative field theories: Jacobi dynamics
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• Noncommutative Supergeometry and Quantum Supergroups
• Membership deformation of commutativity and obscure $n$-ary algebras
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• Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs