A Generalised Hopf Algebra for Solitons

Group factorisation plays a vital part in the inverse scattering procedure [2,10]. For example the Riemann-Hilbert problem is a (not quite exact) factorisation of group valued functions on the real line into functions analytic on the lower half plane times functions analytic on the upper half plane. However there is a problem, a group valued function which is analytic on the lower half plane need not have an inverse which is analytic there. On the Lie algebra level all is well since any smooth loop which is uniformly sufficiently close to the identity and is analytic on the lower half plane has an inverse which is also analytic on the lower half plane. To avoid the problem, we look at the Lie algebras or a neighbourhood of a group near the identity. This corresponds in inverse scattering to looking at solutions not too far from the vacuum. However the soliton solutions for many integrable systems are characterised by meromorphic loops, and there the factors are very definitely not closed under inverse. For example, if we take the meromorphic function given by (for P ∈ Mn(C) a Hermitian projection matrix and P ⊥ = 1− P )