×

zbMATH — the first resource for mathematics

Discretization of Baker-Akhiezer modules and commuting difference operators in several discrete variables. (English. Russian original) Zbl 1351.12001
Trans. Mosc. Math. Soc. 2013, 261-279 (2013); translation from Tr. Mosk. Mat. O.-va 74, No. 2, 317-338 (2013).
Summary: We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are modules over the ring of difference operators, as a discretization of Baker-Akhiezer modules, which are modules over the ring of differential operators. We use it to construct commuting difference operators with matrix coefficients in several discrete variables.

MSC:
12H10 Difference algebra
39A70 Difference operators
47B39 Linear difference operators
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] I. M. Krichever and S. P. Novikov, A two-dimensionalized Toda chain, commuting difference operators, and holomorphic vector bundles, Uspekhi Mat. Nauk 58 (2003), no. 3(351), 51 – 88 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 3, 473 – 510. · Zbl 1060.37068
[2] Igor Moiseevich Krichever, Algebraic curves and nonlinear difference equations, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 215 – 216 (Russian). · Zbl 0382.39003
[3] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equation, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 115 – 153.
[4] A. E. Mironov, Discrete analogues of Dixmier operators, Mat. Sb. 198 (2007), no. 10, 57 – 66 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 9-10, 1433 – 1442. · Zbl 1144.47029
[5] I.M. Krichever, Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surveys, 32:6 (1977), 32:6, 185-213. · Zbl 0386.35002
[6] A.B. Zheglov, On rings of commuting partial differential operators, arXiv:1106.0765 (to appear in St. Petersburg Math. J.).
[7] H. Kurke, D. Osipov and A. Zheglov, Commuting differential operators and higher-dimensional algebraic varieties, arXiv:1211.0976. · Zbl 1306.37077
[8] Atsushi Nakayashiki, Structure of Baker-Akhiezer modules of principally polarized abelian varieties, commuting partial differential operators and associated integrable systems, Duke Math. J. 62 (1991), no. 2, 315 – 358. · Zbl 0732.14008
[9] Atsushi Nakayashiki, Commuting partial differential operators and vector bundles over abelian varieties, Amer. J. Math. 116 (1994), no. 1, 65 – 100. · Zbl 0809.14016
[10] Tetsuji Miwa, On Hirota’s difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 1, 9 – 12. · Zbl 0508.39009
[11] E. Date, M. Jimbo and T. Miwa, Method for generating discrete soliton equations I, J. Phys. Soc. Japan 51-12 (1982), 4116-4124, ibid. II, J. Phys. Soc. Japan 51-12 (1982), 4125-4131, ibid. III, J. Phys. Soc. Japan 52-2 (1983), 388-393, ibid. IV, J. Phys. Soc. Japan 52-3 (1983), 761-765,ibid. V, J. Phys. Soc. Japan 52-3 (1983), 766-771.
[12] Irina A. Melnik and Andrey E. Mironov, Baker-Akhiezer modules on rational varieties, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), Paper 030, 15. · Zbl 1190.14031
[13] Atsushi Nakayashiki, On hyperelliptic Abelian functions of genus 3, J. Geom. Phys. 61 (2011), no. 6, 961 – 985. · Zbl 1215.14032
[14] Felix Klein, Ueber hyperelliptische Sigmafunctionen, Math. Ann. 27 (1886), no. 3, 431 – 464 (German). · JFM 18.0418.02
[15] Felix Klein, Ueber hyperelliptische Sigmafunctionen, Math. Ann. 32 (1888), no. 3, 351 – 380 (German). · JFM 20.0491.01
[16] V. M. Bukhshtaber, D. V. Leĭkin, and V. Z. Ènol\(^{\prime}\)skiĭ, Rational analogues of abelian functions, Funktsional. Anal. i Prilozhen. 33 (1999), no. 2, 1 – 15, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 33 (1999), no. 2, 83 – 94. · Zbl 1056.14049
[17] Atsushi Nakayashiki, On algebraic expressions of sigma functions for (\?,\?) curves, Asian J. Math. 14 (2010), no. 2, 175 – 211. · Zbl 1214.14028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.