Homogeneous Manifolds, Factorisation Problems and Modified KdV Equations
Since the work  of G. Wilson the study of commuting flows in Lie groups has become an important tool in the theory of integrable systems. Our treatment of that subject can be summarised as follows. In a given Lie group G we can construct a set of commuting flows defined as the left translations by the elements of an abelian subgroup of G. Then, if one is able to define a subgroup G + of the group G for wich the homogeneous manifold G/G+ is locally isomorphic to a subgroup G_ ⊂ G, the differential description of the projection of the above mentioned flows in the Lie algebra of G_ is given by the integrable equations. The use of the classical r-matrix formalism furnishes with a systematic approach to that construction in the context of the associated Lie algebra. See  for details and consequences of this construction for Banach-Lie groups of continous linear operators in Hilbert space. For loop groups, infinite dimensional Lie groups defined as the set of smooth maps on the circle with values in some simple group of finite dimension, one can consider two types of canonically defined abelian subgroups, principal and homogeneous subgroups, and deduce from them equations of the KdV and AKNS type respectively (see  for the AKNS case).
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