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STRICT VERSIONS OF VARIOUS MATRIX HIERARCHIES RELATED TO SLn-LOOPS AND THEIR COMBINATIONS

Gerard Franciscus Helminck

Abstract


Let t be a commutative Lie subalgebra of sln(C) of maximal
dimension. We consider in this paper three spaces of t-loops that each get deformed in a different way. We require that the deformed generators of each of them evolve w.r.t. the commuting flows they generate according to a certain, different set of Lax equations. This leads to three integrable hierarchies: the (sln(C), t)-hierarchy, its strict version and the combined (sln(C), t)-hierarchy. For n = 2 and t the diagonal matrices, the (sl2(C), t)- hierarchy is the AKNS-hierarchy. We treat their interrelations and show that all three have a zero curvature form. Furthermore, we discuss their linearization and we conclude by giving the construction of a large class of
solutions.

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References


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