6533b829fe1ef96bd128abc9

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Integrability Conditions: Recent Results in the Theory of Integrable Models

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This paper reports various results achieved recently in the theory of integrable models. These are summarised in the Fig.1! At the Chester meeting [1] two of the authors were concerned [1] with the local Riemann-Hilbert problem (double-lined box in the centre of Fig.1), its limit as a non-local Riemann-Hilbert problem used to solve classical integrable models in 2+1 dimensions (two space and one time dimensions) [2,3], and the connection of this Riemann-Hilbert problem with Ueno’s [4] Riemann-Hilbert problem associated with the representation of the algebra gl(∞) in terms of Z⊗Z matrices (Z the integers) and the solution of the K-P equations in 2+1. We were also concerned [1] with the construction of the integrable models in 1+1 dimensions from the loop algebras ĝ = g⊗[λ,λ-1] where g is a simple finite dimensional Lie algebra and λ ∈ ℂ. Extensions to super-Lie algebras and super-integrable models in 1+1 were also sketched [1,5,6].