Quantum line bundles on noncommutative sphere

Abstract

Noncommutative (NC) sphere is introduced as a quotient of the enveloping algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere (we call them quantum line bundles) and define a multiplicative structure in their family. Also, we compute a pairing between certain quantum line bundles and finite dimensional representations of the NC sphere in the spirit of the NC index theorem. A new approach to constructing the differential calculus on a NC sphere is suggested. The approach makes use of the projective modules in question and gives rise to a NC de Rham complex being a deformation of the classical one.

LaTeX file, 15 pp, no figures. Some clarifying remarks are added at the beginning of section 2 and into section 5

References in corpus (3)

• Introduction to M(atrix) theory and noncommutative geometry, Part II
• Introduction to M(atrix) theory and noncommutative geometry
• Instanton algebras and quantum 4-spheres

Cited by in corpus (5)

• Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
• Braided Differential Operators on Quantum Algebras
• Cayley-Hamilton theorem for quantum matrix algebras of GL(m|n) type
• Weyl approach to representation theory of reflection equation algebra
• Maxwell operator on q-Minkowski space and q-hyperboloid algebras