Geodesic
Lax operators, Poisson groups, and the~differential Galois theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 181 (2014) no. 1, pp. 173-199

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We examine the transfer of a Poisson structure in a space of differential or difference Lax operators to the space of solutions of the corresponding auxiliary problem (the space of wave functions) in detail. We investigate a spontaneous symmetry breaking resulting in the appearance of a nontrivial Poisson structure on the differential or difference Galois group. We review the difference version of the Drinfeld–Sokolov theory and describe a new type of classical $r$-matrices related to generalized exchange algebras.
Mots-clés : Poisson reduction, Poisson–Lie group
Keywords: classical $r$-matrix, difference operator, differential Galois group, difference Galois group. 
@article{TMF_2014_181_1_a8,
     author = {M. A. Semenov-Tian-Shansky},
     title = {Lax operators, {Poisson} groups, and the~differential {Galois} theory},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {173--199},
     publisher = {mathdoc},
     volume = {181},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2014_181_1_a8/}
}
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M. A. Semenov-Tian-Shansky. Lax operators, Poisson groups, and the~differential Galois theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 181 (2014) no. 1, pp. 173-199. http://geodesic.mathdoc.fr/item/TMF_2014_181_1_a8/