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@article{AA_2013_25_5_a3, author = {A. B. Zheglov}, title = {On rings of commuting partial differential operators}, journal = {Algebra i analiz}, pages = {86--145}, publisher = {mathdoc}, volume = {25}, number = {5}, year = {2013}, language = {ru}, url = {http://geodesic.mathdoc.fr/item/AA_2013_25_5_a3/} }
A. B. Zheglov. On rings of commuting partial differential operators. Algebra i analiz, Tome 25 (2013) no. 5, pp. 86-145. http://geodesic.mathdoc.fr/item/AA_2013_25_5_a3/
[1] Atya M., Makdonald I., Vvedenie v kommutativnuyu algebru, Mir, M., 1972 | MR
[2] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR
[3] Gelfand I. M., Dikii L. A., “Asimptotika rezolventy shturm-liuvillevskikh uravnenii i algebra uravnenii Kortevega–de Friza”, Uspekhi mat. nauk, 30:5 (1975), 67–100 ; “Дробные степени операторов и гамильтоновы системы”, Функц. анал. и его прил., 10:4 (1976), 13–29 | MR | Zbl | MR | Zbl
[4] Drinfeld V., “O kommutativnykh podkoltsakh nekotorykh nekommutativnykh kolets”, Funkts. anal. i ego pril., 11:1 (1977), 11–14 | MR | Zbl
[5] Zheglov A. B., Mironov A. E., “Moduli Beikera–Akhiezera, puchki Krichevera i kommutativnye koltsa differentsialnykh operatorov v chastnykh proizvodnykh”, Dalnevost. mat. zh., 12:1 (2012), 20–34 | MR
[6] Zheglov A. B., Osipov D. V., “O nekotorykh voprosakh, svyazannykh s sootvetstviem Krichevera”, Mat. zametki, 81:4 (2007), 528–539 | DOI | MR | Zbl
[7] Krichever I. M., “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, Uspekhi mat. nauk, 32:6 (1977), 183–208 | MR | Zbl
[8] Krichever I. M., “Kommutativnye koltsa obyknovennykh lineinykh differentsialnykh operatorov”, Funkts. anal. i ego pril., 12:3 (1978), 20–31 | MR | Zbl
[9] Mironov A. E., “Kommutativnye koltsa differentsialnykh operatorov, otvechayuschie mnogomernym algebraicheskim mnogoobraziyam”, Sib. mat. zh., 43:5 (2002), 1102–1114 | MR | Zbl
[10] Osipov D. V., “Sootvetstvie Krichevera dlya algebraicheskikh mnogoobrazii”, Izv. RAN. Ser. mat., 65:5 (2001), 91–128 | DOI | MR | Zbl
[11] Parshin A. N., “O koltse formalnykh psevdodifferentsialnykh operatorov”, Tr. Mat. in-ta RAN, 224, 1999, 291–305 | MR | Zbl
[12] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR
[13] Berest Yu., Etingof P., Ginzburg V., “Cherednik algebras and differential operators on quasi-invariants”, Duke Math. J., 18:2 (2003), 279–337 | MR
[14] Braverman A., Etingof P., Gaitsgory D., “Quantum integrable systems and differential Galois theory”, Transform. Groups, 2:1 (1997), 31–56 | DOI | MR | Zbl
[15] Etingof P., Ginzburg V., “On $m$-quasi-invariants of a Coxeter group”, Mosc. Math. J., 2:3 (2002), 555–566 | MR | Zbl
[16] Burchnall J. L., Chaundy T. W., “Commutative ordinary differential operators”, Proc. London Math. Soc. Ser. 2, 21 (1923), 420–440 ; Proc. Royal Soc. London Ser. A, 118 (1928), 557–583 | DOI | MR | Zbl | DOI | Zbl
[17] Chalykh O., “Algebro-geometric Schrödinger operators in many dimensions”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 947–971 | DOI | MR | Zbl
[18] Feigin M., Veselov A. P., “Quasi-invariants of Coxeter groups and $m$-harmonic polynomials”, Intern. Math. Res. Not., 2002:10 (2002), 521–545 | DOI | MR | Zbl
[19] Feigin M., Veselov A. P., “Quasi-invariants and quantum integrals of deformed Calogero–Moser systems”, Intern. Math. Res. Not., 2003:46 (2003), 2487–2511 | DOI | MR | Zbl
[20] Grothendieck A., Éléments de géométrie algébrique, v. II, Inst. Hautes Études Sci. Publ. Math., 8, Étude globale élémentoire de guelgues classes de morphismes, 1961, 222 pp.
[21] Kurke H., Osipov D., Zheglov A., “Formal punctured ribbons and two-dimensional local fields”, J. Reine Angew. Math., 629 (2009), 133–170 | MR | Zbl
[22] Kurke H., Osipov D., Zheglov A., “Formal groups arising from formal punctured ribbons”, Intern. J. Math., 21:6 (2010), 755–797 | DOI | MR | Zbl
[23] Kurke H., Osipov D., Zheglov A., Commuting differential operators and higher-dimensional algebraic varieties, Oberwolfach Preprint Ser., No 2, , 2012 http://www.mfo.de/scientific-programme/publications/owp
[24] Kurke H., Osipov D., Zheglov A., Partial differential operators, Sato Grassmanians and non-linear partial differential equations, (to appear)
[25] Mironov A. E., Self-adjoint commuting differential operators and commutative subalgebras of the Weyl algebra, arXiv: 1107.3356[math-ph]
[26] Mokhov O. I., On commutative subalgebras of the Weyl algebra that are related to commuting operators of arbitrary rank and genus, arXiv: 1201.5979[math-sp]
[27] Mulase M., “Category of vector bundles on algebraic curves and infinite dimensional Grassmanians”, Intern. J. Math., 1:3 (1990), 293–342 | DOI | MR | Zbl
[28] Mulase M., “Algebraic theory of the KP equations”, Perspectives in Mathematical Physics, Conf. Proc. Lecture Notes Math. Phys., 3, Intern. Press, Cambridge, MA, 1994, 151–217 | MR | Zbl
[29] Mumford D., The red book of varieties and schemes, Lecture Notes in Math., 1358, Springer-Verlag, Berlin, 1999 | DOI | MR | Zbl
[30] Mumford D., “An algebro-geometric constructions of commuting operators and of solutions to the Toda lattice equations, Korteweg de Vries equation and related non-linear equation”, Proc. Intern. Sympos. on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokunija Book Store, Tokyo, 1978, 115–153 | MR
[31] Mumford D., Tata lectures on Theta, v. II, Progress in Math., 43, Jaconian theta functions and differential equations, Birkhäuser Boston, Inc., Boston, MA, 1984 | MR | Zbl
[32] Nakayashiki A., “Commuting partial differential operators and vector bundles over Abelian varieties”, Amer. J. Math., 116:1 (1994), 65–100 | DOI | MR | Zbl
[33] Parshin A. N., “Integrable systems and local fields”, Comm. Algebra, 29:9 (2001), 4157–4181 | DOI | MR | Zbl
[34] Previato E., “Multivariable Burchnall–Chaundy theory”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 1155–1177 | DOI | MR | Zbl
[35] Rothstein M., “Dynamics of the Krichever construction in several variables”, J. Reine Angew. Math., 572 (2004), 111–138 | MR | Zbl
[36] Sato M., “Soliton equations and universal Grassmann manifold”, Kokyuroku, Res. Inst. Math. Sci., Kyoto Univ., 439 (1981), 30–46 | Zbl
[37] Sato M., Noumi M., Soliton equations and universla Grassmann manifold, Sophia Univ. Lec. Notes Ser. in Math., 18, 1984
[38] Segal G., Wilson G., “Loop Groups and Equations of KdV Type”, Inst. Hautes Ètudes Dci. Publ. Math., 61 (1985), 5–65 | DOI | MR | Zbl
[39] Schur I., “Über vertauschbare lineare Differentialausdrc̈ke”, Sitzungsber. der Berliner Math. Gesel., 4 (1905), 2–8 | Zbl
[40] Verdier J.-L., “Equations differentielles algébriques”, Mathematics and Physics (Paris, 1979/1982), Progress in Math., 37, Birkhäuser Boston, Inc., Boston, MA, 1983, 215–236 | MR
[41] Wallenberg G., “Über die Vertauschbarkeit homogener linearer Differentialausdrücke”, Archiv Math. Phys. Drittle Reihe, 4 (1903), 252–268 | Zbl
[42] Zariski O., “The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface”, Ann. of Math. (2), 76 (1962), 560–615 | DOI | MR | Zbl
[43] Zheglov A. B., Two dimensional KP systems and their solvability, arXiv: math-ph/0503067v2