Skip to main content
Log in

Torsion Free Sheaves on Weierstrass Cubic Curves and the Classical Yang–Baxter Equation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

This work deals with an algebro–geometric theory of solutions of the classical Yang–Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstraß cubic curves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Adler M., vanMoerbeke P., Vanhaecke P.: Algebraic integrability, Painlevé geometry and Lie algebras, A Series of Modern Surveys in Mathematics 47. Springer, Berlin (2004)

    Book  Google Scholar 

  2. Atiyah M.: Vector bundles over an elliptic curve. Proc. Lond. Math. Soc. (3) 7, 414–452 (1957)

    Article  MathSciNet  Google Scholar 

  3. Avan J., Talon M.: Rational and trigonometric constant nonantisymmetric R–matrices. Phys. Lett. B 241(1), 77–82 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  4. Babelon O., Bernard D., Talon M.: Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  5. Ball J., Vinnikov V.: Zero–pole interpolation for matrix meromorphic functions on a compact Riemann surface and a matrix Fay trisecant identity. Am. J. Math. 121(4), 841–888 (1999)

    Article  MathSciNet  Google Scholar 

  6. Barth,W., Hulek, K., Peters, C., Van de Ven, A.: Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 4, Springer (2004)

  7. Bartocci, C., Bruzzo, U., Hernández Ruipérez, D., Fourier–Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics 276, Birkhäuser (2009)

  8. Beauville A., Laszlo Y.: Un lemme de descente. C. R. Acad. Sci. Paris Sér. I Math. 320(3), 335–340 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Belavin A.: Discrete groups and integrability of quantum systems. Funct. Anal. Appl. 14(4), 18–26 (1980)

    MathSciNet  MATH  Google Scholar 

  10. Belavin A., Drinfeld V.: Solutions of the classical Yang–Baxter equation for simple Lie algebras. Funct. Anal. Appl. 16(3), 159–180 (1983)

    Article  MathSciNet  Google Scholar 

  11. Belavin A., Drinfeld V.: The classical Yang–Baxter equation for simple Lie algebras. Funct. Anal. Appl. 17(3), 69–70 (1983)

    MathSciNet  Google Scholar 

  12. Ben–Zvi, D., Biswas, I.: Theta functions and Szegő kernels, Int.Math. Res. Not. (24), 1305-1340 (2003)

  13. Bodnarchuk, L., Burban, I., Drozd, Yu., Greuel, G.-M.: Vector bundles and torsion free sheaves on degenerations of elliptic curves. In: Global Aspects of Complex Geometry, pp. 83–128. Springer, Berlin (2006)

  14. Bodnarchuk L., Drozd Yu.: Stable vector bundles over cuspidal cubics. Cent. Eur. J. Math. 1(4), 650–660 (2003)

    Article  MathSciNet  Google Scholar 

  15. Bodnarchuk L., Drozd Yu.: One class of wild but brick–tame matrix problems. J. Algebra 323(10), 3004–3019 (2010)

    Article  MathSciNet  Google Scholar 

  16. Bodnarchuk L., Drozd Yu., Greuel G.-M.: Simple vector bundles on plane degenerations of an elliptic curve. Trans. Am. Math. Soc. 364(1), 137–174 (2012)

    Article  MathSciNet  Google Scholar 

  17. Burban I.: Stable vector bundles on a rational curve with one node. Ukraïn Mat. Zh. 55(7), 867–874 (2003)

    MathSciNet  MATH  Google Scholar 

  18. Burban, I.: Abgeleitete Kategorien und Matrixprobleme, PhD Thesis, Kaiserslautern 2003. https://kluedo.ub.uni-kl.de/frontdoor/index/index/year/2003/docId/1434

  19. Burban I., Drozd Yu.: Coherent sheaves on rational curves with simple double points and transversal intersections. Duke Math. J. 121(2), 189–229 (2004)

    Article  MathSciNet  Google Scholar 

  20. Burban, I., Drozd, Yu., Greuel, G.-M.: Vector bundles on singular projective curves. In: Applications of Algebraic Geometry to Coding Theory, Physics andComputation, pp. 1–15. KluwerAcademic Publishers, Amsterdam (2001)

    Google Scholar 

  21. Burban I., Galinat L., Stolin A.: Simple vector bundles on a nodal Weierstraß cubic and quasi– trigonometric solutions of CYBE. J. Phys. A: Math. Theor. 50, 454002 (2017)

    Article  ADS  Google Scholar 

  22. Burban I., Henrich T.: Vector bundles on plane cubic curves and the classical Yang–Baxter equation. J. Eur. Math. Soc. 17(3), 591–644 (2015)

    Article  MathSciNet  Google Scholar 

  23. Burban I., Kreußler B.: Fourier–Mukai transforms and semi–stable sheaves on nodal Weierstraßcubics. J. Reine Angew. Math. 584, 45–82 (2005)

    Article  MathSciNet  Google Scholar 

  24. Burban I., Kreußler B.: On a relative Fourier–Mukai transform on genus one fibrations. Manuscr. Math. 120(3), 283–306 (2006)

    Article  MathSciNet  Google Scholar 

  25. Burban I., Kreußler B.: Derived categories of irreducible projective curves of arithmetic genus one. Compos Math. 142(5), 1231–1262 (2006)

    Article  MathSciNet  Google Scholar 

  26. Burban, I., Kreußler, B.: Vector bundles on degenerations of elliptic curves and Yang–Baxter equations, Memoirs of the AMS 220(1035), (2012)

    Article  MathSciNet  Google Scholar 

  27. Burban, I., Zheglov, A.: Fourier–Mukai transform on Weierstraß cubics and commuting differential operators. arXiv:1602.08694

  28. Chari V., Pressley A.: A Guide to Quantum Groups. Cambridge University Press, Cambrige (1994)

    MATH  Google Scholar 

  29. Cherednik I.: On a method of constructing factorized S–matrices in terms of elementary functions. Teoret. Mat. Fiz. 43(1), 117–119 (1980)

    MathSciNet  Google Scholar 

  30. Cherednik I.: Determination of τ–functions for generalized affine Lie algebras. Funct. Anal. Appl. 17(3), 93–95 (1983)

    MathSciNet  Google Scholar 

  31. Drinfeld V.: Quantum groups. J. Soviet Math. 41(2), 898–915 (1988)

    Article  MathSciNet  Google Scholar 

  32. Drozd Yu., Greuel G.-M.: Tame and wild projective curves and classification of vector bundles. J. Algebra 246(1), 1–54 (2001)

    Article  MathSciNet  Google Scholar 

  33. Etingof P., Schiffmann O.: Lectures on Quantum Groups. International Press, Vienna (2002)

    MATH  Google Scholar 

  34. Faddeev, L., Takhtajan, L.: Hamiltonian methods in the theory of solitons. In: Classics in Mathematics, Springer, Berlin (2007)

  35. Fay, J.: The nonabelian Szegő kernel and theta–divisor. In: Curves, Jacobians and Abelian Varieties, Vol. 136, pp. 171–183, Contemp. Math. Am. Math. Soc. Providence, RI (1992)

  36. Fialowski, A.: An example of formal deformations of Lie algebras. In: Deformation Theory of Algebras and Structures and Applications, Vol. 247, pp. 375–401. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer, Amsterdam (1988)

  37. Ginzburg,V., Kapranov,M., Vasserot, E.:Elliptic Algebras and Equivariant Elliptic Cohomology. arXiv:q-alg/9505012

  38. Golod P.: Hamiltonian systems connected with anisotropic affineLie algebras and higher Landau–Lifshits equations. Dokl. Akad. Nauk Ukrain. SSR Ser. A, No. 5, 6–8 (1984)

    MATH  Google Scholar 

  39. Gómez González E., Plaza Martín F.: Addition formulae for non–abelian theta functions and applications. J. Geom. Phys. 48(2–3), 480–502 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  40. Hartshorne, R.: Residues and duality, Lecture Notes in Math. 20, Springer (1966)

  41. Hurtubise J., Markman E.: Surfaces and the Sklyanin bracket. Commun. Math. Phys. 230(3), 485–502 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  42. Kuroki G., Takebe T.: Twisted Wess–Zumino–Witten models on elliptic curves, Comm. Math. Phys. 190(1), 1–56 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  43. Khoroshkin S., Pop I., Samsonov M., Stolin A., Tolstoy V.: On some Lie bialgebra structures on polynomial algebras and their quantization. Commun. Math. Phys. 282(3), 625–662 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  44. Lekili, Y., Polishchuk, A.: Associative Yang–Baxter equation and Fukaya categories of square–tiled surfaces, arXiv:1608.08992

  45. Maillet J.: Lax equations and quantum groups. Phys. Lett. B 245(3–4), 480–486 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  46. Montaner F., Stolin A., Zelmanov E.: Classification of Lie bialgebras over current algebras. Sel. Math. 16(4), 935–962 (2010)

    Article  MathSciNet  Google Scholar 

  47. Neeman A.: The Grothendieck duality theoremvia Bousfield's techniques and Brown representability. J. Am. Math. Soc. 9(1), 205–236 (1996)

    Article  Google Scholar 

  48. Parshin A.: Integrable systems and local fields. Commun. Algebra 29(9), 4157–4181 (2001)

    Article  MathSciNet  Google Scholar 

  49. Polishchuk A.: Homological mirror symmetry with higher products. AMS/IP Stud. Adv.Math. 23, 247–259 (2001)

    Article  MathSciNet  Google Scholar 

  50. Polishchuk A.: Classical Yang–Baxter equation and the A –constraint.. Adv.Math. 168(1), 56–95 (2002)

    Article  MathSciNet  Google Scholar 

  51. Polishchuk A.: Triple Massey products on curves, Fay’s trisecant identity and tangents to the canonical embedding. Mosc. Math. J. 3(1), 105–121 (2003) 259

    MathSciNet  MATH  Google Scholar 

  52. Polishchuk, A.: Massey products on cycles of projective lines and trigonometric solutions of the Yang–Baxter equations, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 573–617, Progr. Math. 270, Birkhäuser (2009)

  53. Raina, A.: Fay’s matrix identity for vector bundles on a curve. Int. J.Math. 23(12), 1250123, 9 pp (2012)

    Article  MathSciNet  Google Scholar 

  54. Reyman A., Semenov–Tian–Shansky M.: Lie algebras and Lax equations with spectral parameter on an elliptic curve. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 150, 104–118 (1986)

    MATH  Google Scholar 

  55. Reyman A., Semenov–Tian–Shansky M.: Integrable systems II Dynamical Systems VII,. Encycl. Math. Sci. 16, 83–259 (1994)

    Google Scholar 

  56. Hernández Ruipérez D., López Martín A., Sánchez Gómez D., Tejero Prieto C.: Moduli spaces of semistable sheaves on singular genus one curves. Int. Math. Res. Not. no. 23, 4428–4462 (2009)

    MATH  Google Scholar 

  57. Schedler T.: Trigonometric solutions of the associative Yang–Baxter equation. Math. Res. Lett. 10(2- 3), 301–321 (2003)

    Article  MathSciNet  Google Scholar 

  58. Silverman J.: Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics. Vol. 151. Springer, Berlin (1994)

    Book  Google Scholar 

  59. Sklyanin, E.: On the complete integrability of the Landau–Lifshitz equation, preprint LOMI E–3–79 (1979)

  60. Skrypnyk T.: Integrable quantum spin chains, non–skew symmetric r–matrices and quasigraded Lie algebras. J. Geom. Phys. 57(1), 53–67 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  61. Skrypnyk T.: Spin chains in magnetic field, non–skew–symmetric classical r–matrices and BCS–type integrable systems. Nuclear Phys. B 806(3), 504–528 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  62. Skrypnyk T.: Quasi–periodic functions on the torus and sl(n)–elliptic Lie algebra. J. Math. Phys. 53(2), 023502 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  63. Skrypnyk T.: Infinite–dimensional Lie algebras, classical r–matrices, and Lax operators: two approaches. J. Math. Phys. 54(10), 103507 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  64. Smyth D.: Modular compactifications of the space of pointed elliptic curves I. Compos. Math. 147(3), 877–913 (2011)

    Article  MathSciNet  Google Scholar 

  65. Stolin A.: On rational solutions of Yang–Baxter equation for \({\mathfrak{s}\mathfrak{l}(n)}\). Math. Scand. 69(1), 57–80 (1991)

    Article  MathSciNet  Google Scholar 

  66. Stolin A.: On rational solutions of Yang–Baxter equations. Maximal orders in loop algebra. Commun. Math. Phys. 141(3), 533–548 (1991)

    Article  ADS  Google Scholar 

  67. Stolin A.: Rational solutions of the classical Yang–Baxter equation and quasi Frobenius Lie algebras. J. Pure Appl. Algebra 137(3), 285–293 (1999)

    Article  MathSciNet  Google Scholar 

  68. Tate J.: Residues of differentials on curves. Ann. Sci. École Norm. Sup. 4(1), 149–159 (1968)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Igor Burban.

Additional information

Communicated by C. Schweigert

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burban, I., Galinat, L. Torsion Free Sheaves on Weierstrass Cubic Curves and the Classical Yang–Baxter Equation. Commun. Math. Phys. 364, 123–169 (2018). https://doi.org/10.1007/s00220-018-3172-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3172-2

Navigation