Non-commutative toroidal orbifolds


The above picture shows a foliation of the 2-sphere (with the slope 1/E). The algebras of foliations of the sphere are related to C*-algebras of (ir)rational rotation and their fixed-point subalgebras under the flip automorphism.

To find out more about non-commutative orbifolds, read the link below:


Folowing references contain more detailed information:

  • Bratteli, O., Elliott, G. A., Evans, D. E., and Kishimoto, A.: Non-commutative spheres, I. International Journal of Mathematics, 2, 139--166.
  • Bratteli, O. and Kishimoto, A.: Non-commutative spheres, III, Irrational rotations. Communications in Mathematical Physics, 147, 605--624.
  • Bratteli, O., Evans, D. E. and Kishimoto, A.: Crossed products of totally disconnected spaces by Z_2 \times Z_2. Ergodic Theory and Dynamical Systems, 13, 445--484.
  • Elliott, G. A. and Evans, D. E.: The structure of the irrational rotation C*-algebra. Annals of Mathematics, 138, 477--501.
  • Evans, D. E.; Kawahigashi, Y.: `Quantum Symmetries on Operator Algebras', Oxford University Press, 1998

Below are some more pictures:

All the diagrams have been drawn by Soren Winkler.


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