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On rings of commuting partial differential operators. (English. Russian original) Zbl 1325.13025
St. Petersbg. Math. J. 25, No. 5, 775-814 (2014); translation from Algebra Anal. 25, No. 5, 86-145 (2013).
A natural generalization is given for the classification of commutative rings of ordinary differential operators, as presented by Krichever, Mumford, Mulase. The commutative rings of operators in a completed ring of partial differential operators in two variables (satisfying certain mild conditions) are classified in terms of Parshin’s generalized geometric data. This classification involves a generalization of M. Sato’s theory and is constructible both ways.
The paper consists of four sections. Section 1 is an introduction. In Section 2 some known facts about rings of partial differential operators are reviewed, new notations are introduced, the author develops a generalization of M. Sato’s theory. In Section 3 a classification of commutative rings of differential operators is given at first in terms of the generalized Schur pairs. After doing this the author shows that the category of the generalized Schur pairs is equivalent to the category of geometric data (a generalized version of the geometric data from [D. V. Osipov, Izv. Math. 65, No. 5, 941–975 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, No. 5, 91–128 (2001; Zbl 1068.14053)] and [A. N. Parshin, Commun. Algebra 29, No. 9, 4157–4181 (2001; Zbl 1014.14015)]). Section 4 provides some examples. A theorem about algebro-geometric properties of maximal commutative subrings of partial differential operators in two variables is proven here as well. In particular the author shows that such rings are Cohen-Macaulay.

MSC:
13N15 Derivations and commutative rings
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14D15 Formal methods and deformations in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
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References:
[1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601
[2] N. Bourbaki, Éléments de mathématique, Masson, Paris, 1998 (French). Algèbre commutative. Chapitre 10. [Commutative algebra. Chapter 10]; Profondeur, régularité, dualité. [Depth, regularity, duality]. · Zbl 0211.02401
[3] I. M. Gel\(^{\prime}\)fand and L. A. Dikiĭ, Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations, Uspehi Mat. Nauk 30 (1975), no. 5(185), 67 – 100 (Russian).
[4] V. G. Drinfel\(^{\prime}\)d, Commutative subrings of certain noncommutative rings, Funkcional. Anal. i Priložen. 11 (1977), no. 1, 11 – 14, 96 (Russian).
[5] A. B. Zheglov and A. E. Mironov, Baker-Akhiezer modules, Krichever sheaves, and commutative rings of partial differential operators, Dal\(^{\prime}\)nevost. Mat. Zh. 12 (2012), no. 1, 20 – 34 (Russian, with English and Russian summaries). · Zbl 1286.14060
[6] A. B. Zheglov and D. V. Osipov, On some problems associated with the Krichever correspondence, Mat. Zametki 81 (2007), no. 4, 528 – 539 (Russian, with Russian summary); English transl., Math. Notes 81 (2007), no. 3-4, 467 – 476. · Zbl 1134.14023
[7] Igor Moiseevich Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Uspehi Mat. Nauk 32 (1977), no. 6(198), 183 – 208, 287 (Russian).
[8] Igor Moiseevich Krichever, Commutative rings of ordinary linear differential operators, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 20 – 31, 96 (Russian). · Zbl 0408.34008
[9] A. E. Mironov, Commutative rings of differential operators corresponding to multidimensional algebraic varieties, Sibirsk. Mat. Zh. 43 (2002), no. 5, 1102 – 1114 (Russian, with Russian summary); English transl., Siberian Math. J. 43 (2002), no. 5, 888 – 898. · Zbl 1006.14016
[10] D. V. Osipov, The Krichever correspondence for algebraic varieties, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 5, 91 – 128 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 5, 941 – 975. · Zbl 1068.14053
[11] A. N. Parshin, On a ring of formal pseudo-differential operators, Tr. Mat. Inst. Steklova 224 (1999), no. Algebra. Topol. Differ. Uravn. i ikh Prilozh., 291 – 305 (Russian); English transl., Proc. Steklov Inst. Math. 1(224) (1999), 266 – 280. · Zbl 1008.37042
[12] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[13] Yuri Berest, Pavel Etingof, and Victor Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), no. 2, 279 – 337. · Zbl 1067.16047
[14] A. Braverman, P. Etingof, and D. Gaitsgory, Quantum integrable systems and differential Galois theory, Transform. Groups 2 (1997), no. 1, 31 – 56. · Zbl 0901.58021
[15] Pavel Etingof and Victor Ginzburg, On \?-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002), no. 3, 555 – 566. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. · Zbl 1028.81027
[16] J. L. Burchnall and T. W. Chaundy, Commutative Ordinary Differential Operators, Proc. London Math. Soc. S2-21, no. 1, 420. · JFM 49.0311.03
[17] Oleg Chalykh, Algebro-geometric Schrödinger operators in many dimensions, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), no. 1867, 947 – 971. · Zbl 1153.14303
[18] M. Feigin and A. P. Veselov, Quasi-invariants of Coxeter groups and \?-harmonic polynomials, Int. Math. Res. Not. 10 (2002), 521 – 545. · Zbl 1009.20044
[19] M. Feigin and A. P. Veselov, Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems, Int. Math. Res. Not. 46 (2003), 2487 – 2511. · Zbl 1034.81024
[20] A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222 (French).
[21] Herbert Kurke, Denis Osipov, and Alexander Zheglov, Formal punctured ribbons and two-dimensional local fields, J. Reine Angew. Math. 629 (2009), 133 – 170. · Zbl 1168.14002
[22] Herbert Kurke, Denis V. Osipov, and Alexander B. Zheglov, Formal groups arising from formal punctured ribbons, Internat. J. Math. 21 (2010), no. 6, 755 – 797. · Zbl 1203.14012
[23] H. Kurke, D. V. Osipov, and A. B. Zheglov, Commuting differential operators and higher-dimensional algebraic varieties, Oberwolfach Preprint Ser., no. 2, 2012, http://www.mfo.de/scientific-programme/publications/owp. · Zbl 1306.37077
[24] H. Kurke, D. V. Osipov, and A. B. Zheglov, Partial differential operators, Sato Grassmanians and non-linear partial differential equations (to appear). · Zbl 1306.37077
[25] A. E. Mironov, Self-adjoint commuting differential operators and commutative subalgebras of the Weyl algebra, arXiv:math-ph/1107.3356.
[26] O. I. Mokhov, On commutative subalgebras of the Weyl algebra that are related to commuting operators of arbitrary rank and genus, arXiv:math-sp/1201.5979. · Zbl 1347.47027
[27] Motohico Mulase, Category of vector bundles on algebraic curves and infinite-dimensional Grassmannians, Internat. J. Math. 1 (1990), no. 3, 293 – 342. · Zbl 0723.14010
[28] Motohico Mulase, Algebraic theory of the KP equations, Perspectives in mathematical physics, Conf. Proc. Lecture Notes Math. Phys., III, Int. Press, Cambridge, MA, 1994, pp. 151 – 217. · Zbl 0837.35132
[29] David Mumford, The red book of varieties and schemes, Second, expanded edition, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1999. Includes the Michigan lectures (1974) on curves and their Jacobians; With contributions by Enrico Arbarello. · Zbl 0945.14001
[30] 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.
[31] David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. · Zbl 0549.14014
[32] Atsushi Nakayashiki, Commuting partial differential operators and vector bundles over abelian varieties, Amer. J. Math. 116 (1994), no. 1, 65 – 100. · Zbl 0809.14016
[33] A. N. Parshin, Integrable systems and local fields, Comm. Algebra 29 (2001), no. 9, 4157 – 4181. Special issue dedicated to Alexei Ivanovich Kostrikin. · Zbl 1014.14015
[34] Emma Previato, Multivariable Burchnall-Chaundy theory, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), no. 1867, 1155 – 1177. · Zbl 1153.37420
[35] Mitchell Rothstein, Dynamics of the Krichever construction in several variables, J. Reine Angew. Math. 572 (2004), 111 – 138. · Zbl 1142.37362
[36] M. Sato, Soliton equations as dynamical systems on an infinity dimensional Grassmann manifold, Res. Inst. Math. Sci. 439 (1981), 30-46.
[37] M. Sato and M. Noumi, Soliton equations and universal Grassmann manifold, Sophia Univ. Lec. Notes Ser. in Math., vol. 18, Sophia univ., Sophia, 1984. · Zbl 0541.58001
[38] Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5 – 65. · Zbl 0592.35112
[39] I. Schur, Über vertauschbare lineare Differentialausdrcke, Sitzungsber. der Berliner Math. Gesel. 4 (1905), 2-8. · JFM 36.0387.01
[40] Jean-Louis Verdier, Équations différentielles algébriques, Mathematics and physics (Paris, 1979/1982) Progr. Math., vol. 37, Birkhäuser Boston, Boston, MA, 1983, pp. 215 – 236 (French). · Zbl 0414.14012
[41] G. Wallenberg, Über die Vertauschbarkeit homogener linearer Differentialausdrücke, Archiv Math. Phys., Drittle Reihe 4 (1903), 252-268. · JFM 34.0350.01
[42] Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560 – 615. · Zbl 0124.37001
[43] A. B. Zheglov, Two dimensional KP systems and their solvability, arXiv:math-ph/0503067v2.
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