Lectures on Differential Geometry of Modules and Rings

Abstract

Generalizing differential geometry of smooth vector bundles formulated in algebraic terms of the ring of smooth functions, its derivations and the Koszul connection, one can define differential operators, differential calculus and connections on modules over arbitrary commutative, graded commutative and noncommutative rings. For instance, this is the case of quantum theory, SUSY theory and noncommutative geometry, respectively. The relevant material on this subject is summarized.

137 pages

References in corpus (12)

• Gravity coupled with matter and foundation of non-commutative geometry
• A Short Survey of Noncommutative Geometry
• Connections on central bimodules
• Linear Connections in Non-Commutative Geometry
• Supersymmetric quantum theory and non-commutative geometry
• On Curvature in Noncommutative Geometry
• Vector Fields on Quantum Groups
• Spectral triples and associated Connes-de Rham complex for the quantum SU(2) and the quantum sphere
• Quantum mechanics with time-dependent parameters
• Principal Bundles and the Dixmier Douady Class
• Fibre Bundles, Jet Manifolds and Lagrangian Theory. Lectures for Theoreticians
• A dilemma of nonequivalent definitions of differential operators in noncommutative geometry

Cited by in corpus (5)

• Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials
• A Supermanifold structure on Spaces of Morphisms between Supermanifolds
• Differential operators on Lie and graded Lie algebras
• Deformation quantization on jet manifolds
• Differential operators on Schwartz distributions. Jet formalism