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Teoreticheskaya i Matematicheskaya Fizika, 2014, Volume 181, Number 1, Pages 173–199
DOI: https://doi.org/10.4213/tmf8729 (Mi tmf8729) 		 
This article is cited in 1 scientific paper (total in 1 paper)

Lax operators, Poisson groups, and the differential Galois theory

M. A. Semenov-Tian-Shanskyab

a St. Petersburg Branch of the Steklov Institute of Mathematics , RAS, St. Petersburg, Russia
b Institut de Mathématiques de Bourgogne, Dijon, France
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DOI: https://doi.org/10.4213/tmf8729
Abstract: 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.
Keywords: Poisson reduction, Poisson–Lie group, classical $r$-matrix, difference operator, differential Galois group, difference Galois group.
Received: 11.06.2014
English version:
Theoretical and Mathematical Physics, 2014, Volume 181, Issue 1, Pages 1279–1301
DOI: https://doi.org/10.1007/s11232-014-0212-8
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: M. A. Semenov-Tian-Shansky, “Lax operators, Poisson groups, and the differential Galois theory”, TMF, 181:1 (2014), 173–199; Theoret. and Math. Phys., 181:1 (2014), 1279–1301
Citation in format AMSBIB
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    • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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