Mathematics – Differential Geometry
Scientific paper
2008-05-14
J. Geom. Phys. 58 (2008) 1792-1800
Mathematics
Differential Geometry
12 Pages
Scientific paper
10.1016/j.geomphys.2008.09.003
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.
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