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Structure and duality of \(\mathcal D\)-modules related to KP hierarchy. (English) Zbl 0764.58014

The authors give a global vector field naturally defined over the universal Grassmannian manifold (UGM) of Sato [see M. Sato, Proc. Symp. Pure Math. 49, Pt. 1, 51-66 (1989; Zbl 0688.58016)] and M. Sato and Y. Sato [Nonlinear partial differential equations in applied science, Proc. U.S.-Jap. Semin., Tokyo, 1982, North-Holland Math. Stud. 81, 259-271 (1983; Zbl 0528.58020)]. With this, the UGM becomes a differential-algebraic variety in the sense of Kolchin. The main results in the paper deal with cohomological properties (projectivity and duality) of some \({\mathcal D}\)-modules on UGM arising in the KP hiearchy (see loc. cit.). The paper also contains a differential-algebraic interpretation of the KP hierarchy.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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)
58B99 Infinite-dimensional manifolds
58D25 Equations in function spaces; evolution equations
14K25 Theta functions and abelian varieties
14M15 Grassmannians, Schubert varieties, flag manifolds
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