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Commuting matrix differential operators and loop algebras

✍ Scribed by Masoto Kimura; Pol Vanhaecke

Publisher
Elsevier Science
Year
2001
Tongue
French
Weight
146 KB
Volume
125
Category
Article
ISSN
0007-4497

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✦ Synopsis

We consider for fixed positive integers p and q which are coprime the space of all pairs (P , Q) of commuting matrix differential operators (of a fixed size n), where P is monic of order p and Q is normalized of order q. We use the vector valued Sato Grassmannian to construct a natural bijection to an affine subspace of the loop algebra gl(nq)((Ξ» -1 )). In the scalar case (n = 1) the KP flows on the Grassmannian, which are known to trace out Jacobians, lead to commuting flows on this affine space. These flows are Hamiltonian with respect to a family of Poisson structures which are obtained from a family of Lie brackets on the loop algebra.

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## Dedicated to A. Uhhnann i n h o r a o e c r of his eixtkth birthday and a. La8m.e~ in hollour of hi8 fiftieth birthday By E. SOHOLZ and W. TIMMEBMANN of Dresden