Advertisement

Mathematische Zeitschrift

, Volume 281, Issue 3–4, pp 1061–1087 | Cite as

Introducing supersymmetric frieze patterns and linear difference operators

  • Sophie Morier-Genoud
  • Valentin OvsienkoEmail author
  • Serge Tabachnikov
Article

Abstract

We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hill’s operators. The space of these “superfriezes” is an algebraic supervariety, isomorphic to the space of supersymmetric second order difference equations, called Hill’s equations.

Keywords

Supercommutative algebra Frieze pattern Difference equation Cluster algebra 

Notes

Acknowledgments

The first three authors would like to thank the Centro Internazionale per la Ricerca Matematica, the Mathematics Department of the University of Trento and the foundation Bruno Kessler for excellent conditions they offered us. We are pleased to thank Frederic Chapoton and Dimitry Leites for interesting discussions, special thanks to Dimitry for a careful reading of the first version of this paper. S. M-G. and V. O. are grateful to the Institute for Computational and Experimental Research in Mathematics for its hospitality. S. M-G. and V. O. were partially supported by the PICS05974 “PENTAFRIZ” of CNRS. S. T. was supported by NSF grant DMS-1105442. A. U.’s research was supported by Russian Science Foundation (Project N 14-11-00335).

References

  1. 1.
    Assem, I., Reutenauer, C., Smith, D.: Friezes. Adv. Math. 225, 3134–3165 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Berezin, F.: Introduction to Superanalysis. Mathematical Physics and Applied Mathematics, vol. 9. D. Reidel Publishing Co, Dordrecht (1987)Google Scholar
  3. 3.
    Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81, 595–616 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Conway, J.H., Coxeter, H.S.M.: Triangulated polygons and frieze patterns. Math. Gaz. 57, 87–94 (1973). and 175–183zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Coxeter, H.S.M.: Frieze patterns. Acta Arith. 18, 297–310 (1971)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Deligne, P., Morgan, J.: Notes on Supersymmetry (Following Joseph Bernstein), Quantum Fields and Strings: A Course for Mathematicians, vol. 1, 2 (Princeton, NJ, 1996/1997), pp. 41–97. American Mathematical Society, Providence, RI (1999)Google Scholar
  7. 7.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15, 497–529 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28, 119–144 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154, 63–121 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fomin, S., Zelevinsky, A.: Cluster algebras. IV. Coefficients. Compos. Math. 143, 112–164 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Gauss, C.F.: Pentagramma Mirificum, Werke, Bd. III, pp. 481–490Google Scholar
  12. 12.
    Gauss, C.F.: Pentagramma Mirificum, Werke, Bd VIII, pp. 106–111Google Scholar
  13. 13.
    Gieres, F., Theisen, S.: Superconformally covariant operators and super-\(W\)-algebras. J. Math. Phys. 34, 5964–5985 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. A Foundation for Computer Science. Addison-Wesley Publishing Company, Reading, MA (1994)Google Scholar
  15. 15.
    Grozman, P., Leites, D., Shchepochkina, I.: Lie superalgebras of string theories. Acta Math. Vietnam. 26, 27–63 (2001). http://fr.arxiv.org/abs/hep-th/9702120v1
  16. 16.
    Keller, B.: The periodicity conjecture for pairs of Dynkin diagrams. Ann. Math. 177, 111–170 (2013)zbMATHCrossRefGoogle Scholar
  17. 17.
    Krichever, I.: Commuting difference operators and the combinatorial Gale transform. arXiv:1403.4629
  18. 18.
    Krichever, I., Novikov, S.: A two-dimensionalized Toda chain, commuting difference operators, and holomorphic vector bundles. Russ. Math. Surv. 58, 473–510 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Leites, D.A.: Introduction to the theory of supermanifolds. Russ. Math. Surv. 35, 1–64 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Leites, D. (ed.) Seminar on Supersymmetry (v. 1. Algebra and Calculus: Main Chapters), (J. Bern- stein, D. Leites, V. Molotkov, V. Shander). MCCME, Moscow, 2011, 410 pp (in Russian; a version in English is in preparation but available for perusal)Google Scholar
  21. 21.
    Manin, Yu.: Topics in Noncommutative Geometry. Princeton Univ. Press, Princeton (1991)zbMATHCrossRefGoogle Scholar
  22. 22.
    Manin, Yu.: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften, 289. Springer, Berlin (1988)Google Scholar
  23. 23.
    Muir, T.: A treatise on the Theory of Determinants. Revised and enlarged by William H. Metzler Dover Publishing, New York (1960)Google Scholar
  24. 24.
    Michel, J.-P., Duval, C.: On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative. Int. Math. Res. Not. IMRN, No. 14, 47 pp (2008)Google Scholar
  25. 25.
    Morier-Genoud, S.: Arithmetics of 2-friezes. J. Algebr. Combin. 36, 515–539 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Morier-Genoud, S.: Coxeter’s Frieze Patterns at the Crossroads of Geometry and Combinatorics. To appear in Bulletin of the LMS. arXiv:1503.05049
  27. 27.
    Morier-Genoud, S., Ovsienko, V., Tabachnikov, S.: 2-Frieze patterns and the cluster structure of the space of polygons. Ann. Inst. Fourier 62, 937–987 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Morier-Genoud, S., Ovsienko, V., Schwartz, R., Tabachnikov, S.: Linear difference equations, frieze patterns and combinatorial Gale transform. Forum Math. Sigma 2, e22 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ovsienko, V.: Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities. Oberwolfach DMV-Seminar Band, vol. 31, Birkhaäuser, pp. 231–255 (2001)Google Scholar
  30. 30.
    Ovsienko, V.: Cluster superalgebras. arXiv:1503.01894
  31. 31.
    Ovsienko, V., Schwartz, R., Tabachnikov, S.: The pentagram map: a discrete integrable system. Commun. Math. Phys. 299(2), 409–446 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Radul, A.O.: Superstring Schwartz derivative and the Bott cocycle. In: Kupershmidt, B.A. (ed.) Integrable and Superintegrable Systems, pp. 336–351. World Science Publisher, Teaneck, NJ (1990)Google Scholar
  33. 33.
    Shander, V.N.: Vector fields and differential equations on supermanifolds. Funktsional Anal. Appl. 14, 91–92 (1980)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. http://oeis.org/
  35. 35.
    Ustinov, A.: A short proof of Euler’s identity for continuants. Math. Notes 79, 146–147 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Witten, E.: Notes On Super Riemann Surfaces and Their Moduli. arXiv:1209.2459

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Sophie Morier-Genoud
    • 1
  • Valentin Ovsienko
    • 2
    Email author
  • Serge Tabachnikov
    • 3
    • 4
  1. 1.UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu- Paris Rive Gauche, Case 247Sorbonne UniversitésParisFrance
  2. 2.Laboratoire de Mathématiques U.F.R. Sciences Exactes et NaturellesCNRSREIMS cedex 2France
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  4. 4.ICERMBrown UniversityProvidenceUSA

Personalised recommendations