SU(n)-Connections and Noncommutative Differential Geometry

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.

20 pages, LaTeX2e (use packages amstex, amssymb, theorem, a4, pb-diagram, lamsarrow)

References in corpus (1)

• Submanifolds and Quotient Manifolds in Noncommutative Geometry

Cited by in corpus (4)

• D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds
• Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino term
• More on the admissible condition on differentiable maps $\phi :(X^{Az},E;\nabla )\to Y$ in the construction of the non-Abelian Dirac-Born-Infeld action $S_{DBI}(\phi ,\nabla )$
• Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes