MODIFIED SYSTEMS FOUND BY SYMMETRY REDUCTION ON THE COTANGENT BUNDLE OF A LOOP GROUP

The cotangent bundle T*(G) over tilde of a loop group is considered, not with the canonical symplectic structure, but with a deformation of it engendered by the nontrivial cocycle Sigma: (G) over tilde --> (g) over tilde*. Three symplectic actions on T*(G) over tilde are then considered; the left- and right-actions of (G) over tilde and that of the Diffeomorphism group Diff(S-1). Several examples of systems ''of modified type'' are shown to arise by combining the moment maps for these actions, in conjunction with the r-matrix construction. An abstraction of this idea is discussed and it is shown that the Clebsch integrable case of the motion of a rigid body in an ideal fluid is an example of a system which can be described via precisely the same geometric approach.