Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 3

Projections, modules and connections for the noncommutative cylinder

Pages: 527 – 562



Joakim Arnlind (Department of Mathematics, Linköping University, Linköping, Sweden)

Giovanni Landi (Matematica, Università di Trieste, Italy; Institute for Geometry and Physics (IGAP) Trieste, Italy; and INFN, Trieste, Italy)


We initiate a study of projections and modules over a noncommutative cylinder, a simple example of a noncompact noncommutative manifold. Since its algebraic structure turns out to have many similarities with the noncommutative torus, one can develop several concepts in a close analogy with the latter. In particular, we exhibit a countable number of nontrivial projections in the algebra of the noncommutative cylinder itself, and show that they provide concrete representatives for each class in the corresponding $K_0$ group. We also construct a class of bimodules endowed with connections of constant curvature. Furthermore, with the noncommutative cylinder considered from the perspective of pseudo-Riemannian calculi, we derive an explicit expression for the Levi–Civita connection and compute the Gaussian curvature.

The first-named author is supported by the Swedish Research Council.

The second-named author is partially supported by INFN, Iniziativa Specifica “Gauge and String Theories (GAST)”, and by the “National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM)”, and “Laboratorio Ypatia di Scienze Matematiche (LIA-LYSM)” CNRS - INdAM.

Published 19 August 2020