Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

Request Permissions   Purchase Content 
 
 

 

Matrix divisors on Riemann surfaces and Lax operator algebras


Author: O. K. Sheinman
Translated by:
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2017, 109-121
DOI: https://doi.org/10.1090/mosc/267
Published electronically: December 1, 2017
Full-text PDF

Abstract | References | Additional Information

Abstract: Tyurin parametrization of framed vector bundles is extended to the matrix divisors with an arbitrary semi-simple structure group. The considerations are based on the recently obtained description of Lax operator algebras and finite-dimensional integrable systems in terms of $ \ensuremath {\mathbb{Z}}$-gradings of semi-simple Lie algebras.


References [Enhancements On Off] (What's this?)

  • 1. Atiyah, M.F. Vector bundles over an elliptic curve, Proc. London Math. Soc., V. 7 (1957), 414-452. MR 0131423
  • 2. Biswas, I., Dhillon, A., Hurtubise, J., Wentworth, R. A. A symplectic analog of the Quot scheme. C. R. Math. Acad. Sci. Paris 353 (2015), no. 11, 995-999. MR 3419849
  • 3. Grothendieck, A. Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121-138. MR 0087176
  • 4. Faltings, G. Stable $ G$-bundles and projective connections, J. Algebraic Geometry, V. 2 (1993), 507-568. MR 1211997
  • 5. Feigin, B.L. Integrable systems, shaffle algebras and the Bethe equations. Trans. Moscow Math. Soc. 2016, 203-246. MR 3643971
  • 6. Hitchin, N. Stable bundles and integrable systems, Duke Math. J. 54:1 (1987), 91-114. MR 885778
  • 7. Hitchin, N. The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3), 55(3):1, 1987, 59-126. MR 887284
  • 8. Krichever, I.M., Novikov, S.P. Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili (KP) equation. I. Funct. Anal. and Appl. , 4, 12 (1978), 276-286. MR 0515628
  • 9. Krichever, I.M., Novikov, S.P. Holomorphic bundles over algebraic curves and non-linear equations. Russ. Math. Surv., 6, 35 (1980), 53-79. MR 601756
  • 10. Krichever, I.M. Vector bundles and Lax equations on algebraic curves. Comm. Math. Phys. 229, 229-269 (2002). arXiv:Hep-th/0108110. MR 1923174
  • 11. Krichever, I.M., Sheinman, O.K. Lax operator algebras. Funct. Anal. i Prilozhen., 41 (2007), no. 4, p. 46-59. arXiv:math.RT/0701648. MR 2411605
  • 12. Levin, A., Olshanetsky, M., Smirnov, A., Zotov, A. Characteristic classes and integrable systems. General construction, arXiv:1006.0702.
  • 13. Mehta, V.B., Seshadri, C.S. Moduli of vector bundles on curves with parabolic structures. Math. Ann., V. 248, p. 205-239 (1980). MR 575939
  • 14. Narasimhan, M.S., Seshadri, C.S. Stable and unitary vector bundles on a compact Riemann surface. Ann. Math., V. 82, 540-564, 1965. MR 0184252
  • 15. Seshadri, C.S. Moduli of vector bundles on curves with parabolic structures, Bull. Amer. Math. Soc. 83:1 (1977), 124-126. MR 0570987
  • 16. Steinberg, R. Lectures on Chevalley groups. Yale University, 1967. MR 0466335
  • 17. Sheinman, O.K. Current algebras on Riemann surfaces, De Gruyter Expositions in Mathematics, 58, Walter de Gruyter GmbH & Co. KG, Berlin-Boston, 2012, ISBN: 978-3-11-026452-4, 150 pp. MR 2985911
  • 18. Sheinman, O.K. Lax operator algebras and gradings on semi-simple Lie algebras. Doklady Mathematics, V. 461, no. 2, 2015, 143-145. MR 3442778
  • 19. Sheinman, O.K. Lax operator algebras and gradings on semi-simple Lie algebras. Transform. Groups, Vol. 21, No. 1, 2016, 181-196. DOI 10.1007/s00031-015-9340-y. MR 3459709
  • 20. Sheinman, O.K. Hierarchies of finite-dimensional Lax equations with a spectral parameter on a Riemann surface, and semi-simple Lie algebras. Theoret. and Math. Phys., 185:3 (2015), 1816-1831. MR 3438634
  • 21. Sheinman, O.K. Semi-simple Lie algebras and Hamiltonian theory of finite-dimensional Lax equations with the spectral parameter on a Riemann surface, Contemp. Problems of Math. Mech. and Math. Phys. Proc. Steklov Inst. Math., M:MAIK, 2015, V.290, P.178-188. MR 3488791
  • 22. Sheinman, O.K. Global current algebras and localization on Riemann surfaces, Moscow Math. Journ., V. 15, n. 4, (2015), p. 833-846. MR 3438837
  • 23. Sheinman, O.K. Lax operator algebras and integrable systems. Russian Math. Surveys, 71:1 (2016), 109-156. MR 3507465
  • 24. Tyurin, A.N. On the classification of rank 2 vector bundles over an algebraic curve of arbitrary genus, Izv. AN SSSR. Ser. Math. (1964) 28:1 p. 21-52. Also published in: Tyurin, A.N., Vector Bundles, Collected Works, V.1, F. Bogomolov, A. Gorodentsev, V. Pidstrigach, M. Reid, and N. Tyurin eds., Göttingen, 2008. MR 2742585
  • 25. Tyurin, A.N. Classification of vector bundles over an arbitrary genus algebraic curve. Amer. Math. Soc. Transl. Ser. 2, 63 (1967), 245-279.
  • 26. Tyurin, A.N. On classification of rank $ n$ vector bundles over an arbitrary genus algebraic curve. Amer. Math. Soc. Transl. Ser. 2, 73 (1968), 196-211.
  • 27. Tyurin, A.N. On the classifications of rank 2 vector bundles on an algebraic curve of an arbitrary genus. Izv. AN SSSR, ser. math. 28:1 (1964), 21-52 (see also Collected papers).
  • 28. Tyurin, A.N. Geometry of modules of vector bundles, Russ Math. Surv., 29:6(180) (1974), 59-88 (see also Collected papers). MR 0404261
  • 29. Vinberg, E.B., Gorbatsevich, V.V., Onischik A.L. Structure of Lie groups and Lie algebras, Itogi nauki i techniki, Ser.: Contemp. problems of math. Fundamental directions. M: VINITI, 1990. V. 41, P. 5-258. MR 1056486
  • 30. Weil, A. Généralisation des fonctions abeliennes, J. Math. Pures et Appl., v.17 (1938), 47-87.


Additional Information

O. K. Sheinman
Affiliation: Steklov Mathematical Institute of Russian Academy of Science

DOI: https://doi.org/10.1090/mosc/267
Published electronically: December 1, 2017
Additional Notes: Partial support by the International Research Project GEOMQ11 of the University of Luxembourg and by the OPEN scheme of the Fonds National de la Recherche (FNR), Luxembourg, project QUANTMOD O13/570706, is gratefully acknowledged.
Dedicated: Dedicated to E. B. Vinberg on the occasion of his 80th birthday
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society