Projections, modules and connections for the noncommutative cylinder

• Published in: Advances in theoretical and mathematical physics 2020, Vol.24 (3), p.527-562
• Main Authors: Arnlind, Joakim, Landi, Giovanni
• Format: Article
• Language: English
• ISSN: 1095-0761; 1095-0753; 1095-0753
• DOI: 10.4310/ATMP.2020.v24.n3.a1

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.