Loop groups, discrete versions of some classical integrable systems and rank 2 extensions / Percy Deift, Luen-Chau Li, Carlos Tomei

In 1989, J. Moser and A. P. Veselov considered a class of discrete systems which arise as the Euler-Lagrange equations for a formal sum (1) S=∑ k∈ℤ L(X k ,X k+1 ), where the X k are points on a manifold M n , L(·,·) is a function on Q 2n =M n ×M n , and k∈Z plays the role of the discrete time. The Euler-Lagrange equation gives rise to a map Ψ:(X k ,X k+1 )→(X k+1 ,X k+2 ) which is symplectic with respect to the structure. Particularly, they considered the case of M n =0(N), n=N(N-1)/2 and L(X,Y)=trXJY T , where J is a positive symmetric matrix and discovered that the previous Euler-Lagrange equations can be solved by a QR-type algorithm. The main task of this paper is to give a Lie- algebraic interpretation of the results of Moser and Veselov, in terms of a loop group framework. In this framework the previous discrete systems appear as time-one maps of integrable Hamiltonian flows on coadjoint orbits of appropriate loop groups, which are in turn constructed from more primitive loop groups by means of classical R-matrix theory. Examples in this work include the discrete Euler-Arnold top and the billiard ball problem in an elliptical region in n dimensions. Earlier results of Moser on rank 2 extensions of a fixed matrix can be incorporated into this framework, this implies that many well-known integrable systems, such as the Neumann system, periodic Toda, geodesic flow on an ellipsoid, etc., can also be analyzed by this method. This work would be suitable as a graduate textbook in the modern theory of integrable systems. (Zentralblatt)

Bibliogr. p. 99-101