Geodesic
The~relative frame bundle of an~infinite-dimensional flag variety and solutions of integrable hierarchies
Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 3, pp. 440-471

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We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular $\mathbb Z{\times}\mathbb Z$ matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker–Akhiezer function.
Keywords: upper-triangular $\mathbb Z{\times}\mathbb Z$ matrices, zero curvature form.
Mots-clés : Lax equations 
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     author = {G. F. Helminck and A. G. Helminck and A. V. Opimakh},
     title = {The~relative frame bundle of an~infinite-dimensional flag variety and solutions of integrable hierarchies},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {440--471},
     publisher = {mathdoc},
     volume = {165},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_165_3_a3/}
}
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G. F. Helminck; A. G. Helminck; A. V. Opimakh. The~relative frame bundle of an~infinite-dimensional flag variety and solutions of integrable hierarchies. Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 3, pp. 440-471. http://geodesic.mathdoc.fr/item/TMF_2010_165_3_a3/