## Algebraic Aspects of Integrable Systems: In Memory of Irene by Mark S. Alber, Gregory G. Luther (auth.), A. S. Fokas, I. M. PDF

By Mark S. Alber, Gregory G. Luther (auth.), A. S. Fokas, I. M. Gelfand (eds.)

ISBN-10: 1461224349

ISBN-13: 9781461224341

ISBN-10: 1461275350

ISBN-13: 9781461275350

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5). 7) their behavior under the action of 8L(2, q. Of course there is now a problem because the quantity j-W or (cz + d)2w is not uniquely defined for w non-integral. This can be overcome in several ways. In the case when R is a space of functions on the upper half-plane 1i, we replace the group 8L(2, R) by its universal covering, consisting of matrices (~ ~) E SL(2, R) together with a choice of logarithm of cz + d in 1i, and take for r a subgroup of this covering which maps isomorphically onto a discrete co-finite volume subgroup of 8L(2, R).

Applied to the matrix hierarchies, this involves considerable sophistication. The paper is self-contained and does not expect any special prerequisite from a reader. 1. Introduction Integrable systems of differential equations exist not isolated but united in large communities called hierarchies. All equations inside a hierarchy are commuting with each other. The first known hierarchies were generalized Korteweg-de Vries (KdV) hierarchies, one for every natural number n. , [9]). Then an immense Kadomtsev-Petviashvili (KP) hierarchy was found which united all the KdV's.

Is it true that T(p',p) depends only on p' - p ? For example, a direct calculation shows that Am(P) = Am(P') for m = 1,0, -1, -2, whereas where j = 8z' j8z. This means that A3(P) - A3(p') is essentially multiplication by p' - p, if one writes (dz)-l instead of 8 z at the last place of the right hand side. Question. Can T(P',p) be described in terms of derivations and multiplication in r(X, £x)? All of this has a straightforward supersymmetric version. 46 Paula Beazley Cohen, Yuri Manin, and Don Zagier a Complex powers and 'D-modules.

### Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman by Mark S. Alber, Gregory G. Luther (auth.), A. S. Fokas, I. M. Gelfand (eds.)

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