Riemann surfaces and AF -algebras

2007/10/31 by Igor Nikolaev
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Categorical variable #Computer science #Covariant transformation #Functor #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #Quiver #Riemann surface #Space (punctuation) #math.AG #math.OA #msc:14H55 #msc:46L85

paper · pdf · doi:10.1215/20088752-3544893

published as Ann. Funct. Anal. 7, no. 2 (2016), 371-380 · to appear Annals of Functional Analysis. arXiv admin note: substantial text overlap with arXiv:math/0104077

arxiv created 2015/11/04 · openalex publication_date 2016/04/09 · arxiv updated 2016/06/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras.

Citations

• Noncommutative geometry of algebraic curves
• Classification of holomorphic vector bundles on noncommutative two-tori
• Real Multiplication and noncommutative geometry
• On the geometry and dynamics of diffeomorphisms of surfaces
• On the classification of inductive limits of sequences of semisimple finite-dimensional algebras
• Non-commutative compactifications and elliptic curves
• Quasicoherent sheaves on complex noncommutative two-tori