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A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators. (English) Zbl 1101.37048

Summary: We show that commutative rings of formal pseudodifferential operators can be conjugated as subrings in noncommutative Banach algebras of operators in the presence of certain eigenfunctions. Techniques involve those of the Sato Grassmannian as used in the study of the KP hierarchy as well as the geometry of an infinite-dimensional Stiefel bundle with structure modeled on such Banach algebras. Generalizations of this procedure are considered, too.

MSC:

37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
35Q53 KdV equations (Korteweg-de Vries equations)
58B25 Group structures and generalizations on infinite-dimensional manifolds
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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