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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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