Two-Valued Groups, Kummer Varieties, and Integrable Billiards

Research Exposition
  • 5 Downloads

Abstract

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of \(\sigma \)-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were ”virtual”, while in the present case the resulting bundles are effectively realizable.

Keywords

2-valued groups Kummer varieties Hyperelliptic Jacobians Integrable billiards Semi-stable bundles 

Mathematics Subject Classification

20N20 14H40 14H70 

Notes

Acknowledgements

The authors would like to thank the reviewers for their helpful remarks. The research of one of the authors (V. D.) was partially supported by the Serbian Ministry of Education, Science, and Technological Development, Project 174020 Geometry and Topology of Manifolds, Classical Mechanics, and Integrable Dynamical Systems.

References

  1. Arnol’d, V.: Mathematical methods of classical mechanics. Springer, New York (1978)CrossRefGoogle Scholar
  2. Audin, M.: Spinning tops. An introduction to integrable systems, Cambridge studies in advanced mathematics 51 (1999)Google Scholar
  3. Audin, M.: Courbes algebriques et syst‘emes integrables: geodesiques des quadriques. Expo. Math. 12, 193–226 (1994)MATHGoogle Scholar
  4. Baker, H.F.: On the hyperelliptic sigma functions, Am. J. Math. 301–384 (1898)Google Scholar
  5. Buchstaber, V.M., Novikov, S.P.: Formal groups, power systems and Adams operators. Mat. Sb. (N. S) 84(126), 81–118 (1971). (in Russian) MathSciNetMATHGoogle Scholar
  6. Buchstaber, V.M., Rees, E.G.: Multivalued groups, their representations and Hopf algebras. Transform. Groups 2, 325–249 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. Buchstaber, V.M., Veselov, A.P.: Integrable correspondences and algebraic representations of multivalued groups, Internat. Math. Res. Notices, 381–400 (1996)Google Scholar
  8. Buchstaber, V.M.: Functional equations that are associated with addition theorems for elliptic functions, and two valued groups, Uspekhi Math. Nauk 45 (1990) No 3 (273) 185–186 (Russian) English translation. Russian Math. Surveys 45(3), 213–215 (1990)Google Scholar
  9. Buchstaber, V.M.: n-Valued groups: theory and applications. Moscow Math. J. 6, 57–84 (2006)MathSciNetMATHGoogle Scholar
  10. Buchstaber, V.M.: Polynomial dynamical systems and the Korteweg-de Vries equation. Proc. Steklov Inst. Math. 294, 176–200 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. Buchstaber, V.M., Mikhailov, A.V.: Infinite-dimensional lie algebras determined by the space of symmetric squares of hyperelliptic curves. Funct. Anal. Appl. 51(1), 4–27 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. Buchstaber, V.M., Enolskii, V.Z., Leikin, D.V.: Kleinian functions, hyperelliptic Jacobians and applications., Reviews in Mathematics and Math. Physics, Krichever, I.M., Novikov, S.P. Editors, v. 10, part 2, Gordon and Breach, London, 3–120 (1997)Google Scholar
  13. Buchstaber, V.M., Enolskii, V.Z., Leikin, D.V.: Multi-dimensional sigma-functions. arXiv:1208.0990 [math-ph] (2012)
  14. Buchstaber, V.M., Leykin, D.V.: Addition laws on Jacobian varieties of plane algebraic curves. Proc. Steklov Math. Inst. 251(4), 49–120 (2005)Google Scholar
  15. Buchstaber, V.M., Rees, E.G.: The Gelfand map and symmetric products. Selecta Math. (N.S.) 8(4), 523–535 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. Darboux, G.: Principes de géométrie analytique, p. 519. Gauthier-Villars, Paris (1917)MATHGoogle Scholar
  17. Donagi, R.: Group law on the intersection of two quadrics Ann. Sc. Norm. Sup. Pisa, 217–239 (1980)Google Scholar
  18. Dragović, V.: Poncelet-Darboux curves, their complete decomposition and Marden theorem. Internat. Math. Res. Notes 2011, 3502–3523 (2011)MathSciNetMATHGoogle Scholar
  19. Dragović, V.: Multi-valued hyperelliptic continous fractions of generalized Halphen type. Internat. Math. Res. Notices 2009, 1891–1932 (2009)MATHGoogle Scholar
  20. Dragović, V.: Generalization and Geometrization of the Kowalevski top. Comm. Math. Phys. 298(1), 37–64 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. Dragović, V., Kukić, K.: Discriminantly separable polynomials and quad-graphs. J. Geom. Mech. 6(3), 319–333 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. Dragović, V., Kukić, K.: Systems of the Kowalevski type and discriminantly separable polynomials. Regul. Chaotic Dyn. 19(2), 162–184 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. Dragović, V., Kukić, K.: The Sokolov case, integrable Kirchhoff elasticae, and genus 2 theta-functions via discriminantly separable polynomials. Proc. Steklov Math. Inst. 286, 224–239 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. Dragović, V., Radnović, M.: Geometry of integrable billiards and pencils of quadrics. J. Math. Pures Appl. 85, 758–790 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. Dragović, V., Radnović, M.: Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms. Adv. Math. 219, 1577–1607 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. Dragović, V., Radnović, M.: Poncelet porisms and beyond: integrable billiards, hyperelliptic Jacobians and pencils of quadrics, Frontiers in Mathematics, Birkhauser/Springer, 294, ISBN 978-3-0348-0014-3 (2011)Google Scholar
  27. Dragović, V., Radnović, M.: Bicentennial of the Great Poncelet Theorem. Bull. Am. Math. Soc. 51, 373–445 (2014)CrossRefMATHGoogle Scholar
  28. Dragović, V., Radnović, M.: Pseudo-integrable billiards and arithemetic dynamics. J. Modern Dyn. 8(1), 109–132 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. Dragović, V., Radnović, M.: Periods of pseudo-integrable billiards. Arnold Math. J. 1(1), 69–73 (2015).  https://doi.org/10.1007/s40598-014-0004-0 MathSciNetCrossRefMATHGoogle Scholar
  30. Dragović, V., Radnović, M.: Pseudo-integrable billiards and double reflection nets. Russian Math. Surv. 70(1), 3–34 (2015).  https://doi.org/10.4213/rm9648 MathSciNetMATHGoogle Scholar
  31. Dragović, V., Gajić, B.: Some recent generalizations of the classical rigid body systems. Arnold Math. J. 2(4), 511–578 (2016)MathSciNetCrossRefMATHGoogle Scholar
  32. Dubrovin, B.A., Novikov, S.P.: A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry (Russian) Dokl. Akad. Nauk SSSR 219 (1974), 531-534, English translation: Soviet Math. Dokl. 15 (1974), no. 6, 1597–1601 (1975)Google Scholar
  33. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants and multidimensional deteminants. Birhauser, Boston (1994)CrossRefMATHGoogle Scholar
  34. Golubev, V.V.: Lectures on the integration of motion of a heavy rigid body around a fixed point, Gostechizdat, Moscow, 1953 [in Russian] Israel program for scintific literature, English translation (1960)Google Scholar
  35. Griffiths, Ph, Harris, J.: Principles of Algebraic Geometry. Wiley Classics library edition (1994)Google Scholar
  36. Knörrer, H.: Geodesics on the ellipsoid. Invent. Math. 59, 119–143 (1980)MathSciNetCrossRefMATHGoogle Scholar
  37. Kowalevski, S.: Sur la probleme de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12, 177–232 (1889)MathSciNetCrossRefMATHGoogle Scholar
  38. Moser, J., Veselov, A.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139(2), 217–243 (1991)MathSciNetCrossRefMATHGoogle Scholar
  39. Narasimhan, M., Ramanan, S.: Moduli of vector bundles on compact Riemann surfaces. Ann. Math. 2(89), 14–51 (1969)MathSciNetCrossRefMATHGoogle Scholar
  40. Tyurin, A.N.: On intersection of quadrics. Russian Math. Surv. 30, 51–105 (1975)MathSciNetCrossRefMATHGoogle Scholar
  41. Veselov, A.: Growth and integrability in the dynamics of mappings. Commun. Math. Phys. 145, 181–193 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2018

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Department of Mathematical SciencesThe University of Texas at DallasRichardsonUSA
  3. 3.Mathematical Institute SANUBelgradeSerbia

Personalised recommendations