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,
- 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',
Below are some more pictures:
- slope= 1/3
- slope= 1/3
- slope= 1/7
- slope= 1/7
- slope= 1/15
- slope= 2/3
- slope= 1/1
- slope= 1/1
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 13/15
- slope= 13/15
- slope= 5/7
- slope= 5/7
- slope= 1/3
- slope= 1/7
- slope= 0
- slope= 0
- slope= 1/7
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
- slope= 1/E
All the diagrams have been drawn by Soren Winkler.
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