Geodesic
Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 3, pp. 502-514

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We define and study deformations of the structure constants for a certain class of associative noncommutative algebras generated by deformation-driving algebras (DDAs). These deformations are governed by the central system (CS). We study such a CS in the case where the DDA is the algebra of shifts. We present concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations. We show that the theory of Darboux transformations is completely incorporated into the proposed scheme of deformations, at least in the BSQ case.
Keywords: associative algebra, deformation, integrable system. 
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     author = {B. G. Konopelchenko},
     title = {Continuous-discrete integrable equations and {Darboux} transformations as deformations of associative algebras},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {502--514},
     publisher = {mathdoc},
     volume = {159},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_159_3_a15/}
}
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B. G. Konopelchenko. Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 3, pp. 502-514. http://geodesic.mathdoc.fr/item/TMF_2009_159_3_a15/