[Paper Review] SU(n)-Connections and Noncommutative Differential Geometry

Michel Dubois‐Violette, Thierry Masson|ArXiv.org|Dec 27, 1996

Advanced Topics in Algebra3 references4 citations

TL;DR

This paper establishes a noncommutative differential geometric framework for the algebra of endomorphisms of SU(n)-vector bundles by interpreting ordinary connections as noncommutative 1-forms via derivation-based calculus. It shows that the Lie algebra of derivations forms a Lie algebroid and generalizes connections to noncommutative settings, providing a bridge between gauge theory and noncommutative geometry for SU(n) structures.

ABSTRACT

We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.

Motivation & Objective

• To develop a noncommutative differential geometric framework for the endomorphism algebra of SU(n)-vector bundles.
• To reinterpret standard SU(n) connections as noncommutative 1-forms within a derivation-based differential calculus.
• To characterize the Lie algebra of derivations on the endomorphism algebra as a Lie algebroid structure.
• To generalize ordinary connections to noncommutative connections as a natural extension of the derivation-based formalism.
• To unify concepts from gauge theory and noncommutative geometry in the context of SU(n) bundles.

Proposed method

• Utilizes the algebra of endomorphisms of SU(n)-vector bundles as the noncommutative algebraic foundation.
• Applies a differential calculus based on derivations to define noncommutative differential forms.
• Identifies ordinary SU(n) connections with noncommutative 1-forms in this derivation-based framework.
• Constructs the Lie algebra of derivations on the endomorphism algebra and proves it forms a Lie algebroid.
• Generalizes the notion of connections to noncommutative settings by extending the derivation-based formalism.
• Employs tools from differential geometry and noncommutative geometry, including Lie algebroid theory and endomorphism algebras.

Experimental results

Research questions

• RQ1 How can ordinary SU(n) connections on vector bundles be reformulated within a noncommutative differential geometric framework?
• RQ2 What is the role of the Lie algebra of derivations on the endomorphism algebra of an SU(n)-bundle in noncommutative geometry?
• RQ3 Can the derivation-based differential calculus on endomorphism algebras naturally encode standard gauge connections as noncommutative 1-forms?
• RQ4 How does the structure of a Lie algebroid emerge from the derivations of the endomorphism algebra of an SU(n)-bundle?
• RQ5 What is the generalization of connections in this noncommutative setting, and how does it extend the classical notion?

Key findings

• Ordinary SU(n) connections on vector bundles are naturally identified as noncommutative 1-forms in the derivation-based differential calculus on the endomorphism algebra.
• The Lie algebra of derivations on the endomorphism algebra of an SU(n)-bundle is shown to form a Lie algebroid.
• The derivation-based differential calculus provides a consistent noncommutative framework for gauge-theoretic structures.
• Noncommutative connections generalize classical connections by extending the derivation-based formalism to non-abelian settings.
• The construction establishes a direct correspondence between standard gauge connections and noncommutative differential forms in the SU(n) context.
• The framework provides a geometric interpretation of SU(n) gauge theory within noncommutative differential geometry, enabling new algebraic and geometric insights.