Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 13 (2018), 95 -- 105

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

ON TEA, DONUTS AND NON-COMMUTATIVE GEOMETRY

Igor V. Nikolaev

Abstract. As many will agree, it feels good to complement a cup of tea by a donut or two. This sweet relationship is also a guiding principle of non-commutative geometry known as Serre Theorem. We explain the algebra behind this theorem and prove that elliptic curves are complementary to the so-called non-commutative tori.

2010 Mathematics Subject Classification: 14H52; 46L85
Keywords: Elliptic curve; Non-commutative torus

Full text

References

  1. M.~Auslander, A functorial approach to representation theory, in: Representations of algebras (Puebla, 1980), pp. 105-179, Lecture Notes in Math. 944, Springer, Berlin-New York, 1982. MR0672116(83m:16027). Zbl 0486.16027.

  2. B.~L.~Feigin and A.~V.~Odesskii, Sklyanin's elliptic algebras, Functional Anal. Appl. 23 (1989), 207-214. MR1026987(91e:16037). Zbl 0713.17009.

  3. B.~H.~Gross, Arithmetic on Elliptic Curves with Complex Multiplication, Lecture Notes Math. 776 (1980), Springer. MR0563921(81f:10041). Zbl 0433.14032.

  4. I.~V.~Nikolaev, Noncommutative Geometry, De Gruyter Studies in Mathematics 66, De Gruyter, Berlin, 2017. Zbl 06790762.

  5. M.~A.~Rieffel, Non-commutative tori -- a case study of non-commutative differentiable manifolds, Contemp. Math. 105 (1990), 191-211. MR1047281(91d:58012). Zbl 0713.46046.

  6. J.~P.~Serre, Fasceaux algébriques cohérents, Ann. of Math. 61 (1955), 197-278. MR0068874(16,953c). Zbl 0067.16201.

  7. J.~H.~Silverman and J.~Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR1171452(93g:11003). Zbl 0752.14034.

  8. E.~K.~Sklyanin, Some algebraic structures connected to the Yang--Baxter equations, Functional Anal. Appl. 16 (1982), 27-34. MR0684124(84c:82004).

  9. S.~P.~Smith and J.~T.~Stafford, Regularity of the four dimensional Sklyanin algebra, Compositio Math. 83 (1992), 259-289. MR1175941(93h:16037). Zbl 0758.16001.

  10. A.~B.~Verevkin, On a noncommutative analogue of the category of coherent sheaves on a projective scheme, in Algebra and analysis (Tomsk, 1989), 41-53; Amer. Math. Soc. Transl. Ser. 2, 151, Amer. Math. Soc., Providence, RI, 1992 MR1191171(93j:14002). Zbl 0920.14001.



Igor V. Nikolaev
Department of Mathematics and Computer Science
St. John's University, 8000 Utopia Parkway,
New York, NY 11439, United States.
e-mail: igor.v.nikolaev@gmail.com

http://www.utgjiu.ro/math/sma