On a desingularization of the moduli space of noncommutative tori
HTML articles powered by AMS MathViewer
- by Igor Nikolaev PDF
- Proc. Amer. Math. Soc. 136 (2008), 3769-3774 Request permission
Abstract:
It is shown that the moduli space of the noncommutative tori ${\mathbb A}_{\theta }$ admits a natural desingularization by the group $Ext~ ({\mathbb A}_{\theta },{\mathbb A}_{\theta })$. Namely, we prove that the moduli space of pairs $({\mathbb A}_{\theta }, Ext~ ({\mathbb A}_{\theta },{\mathbb A}_{\theta }))$ is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.References
- Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
- Edward G. Effros, Dimensions and $C^{\ast }$-algebras, CBMS Regional Conference Series in Mathematics, vol. 46, Conference Board of the Mathematical Sciences, Washington, D.C., 1981. MR 623762, DOI 10.1090/cbms/046
- Edward G. Effros and Chao Liang Shen, Approximately finite $C^{\ast }$-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), no. 2, 191–204. MR 563206, DOI 10.1512/iumj.1980.29.29013
- K. R. Goodearl, Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs, vol. 20, American Mathematical Society, Providence, RI, 1986. MR 845783, DOI 10.1090/surv/020
- David Handelman, Extensions for AF $C^{\ast }$ algebras and dimension groups, Trans. Amer. Math. Soc. 271 (1982), no. 2, 537–573. MR 654850, DOI 10.1090/S0002-9947-1982-0654850-0
- John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, DOI 10.1007/BF02395062
- Yu. I. Manin, Real multiplication and noncommutative geometry (ein Alterstraum), The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 685–727. MR 2077591
- M. Pimsner and D. Voiculescu, Imbedding the irrational rotation $C^{\ast }$-algebra into an AF-algebra, J. Operator Theory 4 (1980), no. 2, 201–210. MR 595412
- Marc A. Rieffel, $C^{\ast }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572, DOI 10.2140/pjm.1981.93.415
- Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
Additional Information
- Igor Nikolaev
- Affiliation: The Fields Institute for Mathematical Sciences, Toronto, Ontario, Canada
- Email: igor.v.nikolaev@gmail.com
- Received by editor(s): March 30, 2007
- Received by editor(s) in revised form: September 6, 2007
- Published electronically: June 24, 2008
- Additional Notes: The author was partially supported by NSERC
- Communicated by: Marius Junge
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3769-3774
- MSC (2000): Primary 14H52, 46L85
- DOI: https://doi.org/10.1090/S0002-9939-08-09465-3
- MathSciNet review: 2425714