Loop group decompositions in almost split real forms and applications to soliton theory and geometry

We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action-by the whole subgroup of loops which extend holomorphically to the exterior disc-on the U-hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps (1+1 wave maps) from surfaces into compact symmetric spaces. An “Iwasawa-type” decomposition of the same type of real form, with respect to a fixed point subgroup of an involution of the second kind, is also proved, and an application given.