Elliptic Integrable Systems: A Comprehensive Geometric Interpretation
Idrisse Khemar
Jan 2012 · American Mathematical Soc.
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In this paper, we study all the elliptic integrable systems, in the sense of C. L. Terng [66], that is to say, the family of all the m-th elliptic integrable systems associated to a k′-symmetric space N=G/G0. Here m∈N and k′∈N∗ are integers. For example, it is known that the first elliptic integrable system associated to a symmetric space (resp. to a Lie group) is the equation for harmonic maps into this symmetric space (resp. this Lie group). Indeed it is well known that this harmonic maps equation can be written as a zero curvature equation: dαλ + 12[αλ∧αλ]=0, ∀λ∈C∗, where αλ = λ−1α1′+α0+λα1′′ is a 1-form on a Riemann surface L taking values in the Lie algebra g. This 1-form αλ is obtained as follows. Let f:L→N=G/G0 be a map from the Riemann surface L into the symmetric space G/G0. Then let F:L→G be a lift of f, and consider α=F−1. dF its Maurer-Cartan form. Then decompose α according to the symmetric decomposition g = g0 ⊕ g1 of g : α=α0+α1. Finally, we define αλ: = λ−1α1′+α0+λα1′′, ∀λ∈C∗, where α1′,α1′′ are the resp. (1,0) and (0,1) parts of α1. Then the zero curvature equation for this αλ, for all λ∈C∗, is equivalent to the harmonic maps equation for f:L→N=G/G0, and is by definition the first elliptic integrable system associated to the symmetric space G/G0. Thus the methods of integrable system theory apply to give generalised Weierstrass representations, algebro-geometric solutions, spectral deformations, and so on. In particular, we can apply the DPW method [23] to obtain a generalised Weierstrass representation. More precisely, we have a Maurer-Cartan equation in some loop Lie algebra Λgτ = {ξ:S1→g|ξ(−λ) = τ(ξ(λ))}.
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About the author
Idrisse Khemar is at Universite Henri Poincare, Vandoeuvre-les-Nancy,France.
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