Abstract
We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.
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With best wishes to Igor Viktorovich Tyutin
This research was performed at the Steklov Mathematical Institute of Russian Academy of Sciences and was funded by a grant from the Russian Science Foundation (Project No. 14-50-00005).
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 187, No. 3, pp. 433–446, June, 2016.
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Pogrebkov, A.K. Commutator identities on associative algebras, the non-Abelian Hirota difference equation and its reductions. Theor Math Phys 187, 823–834 (2016). https://doi.org/10.1134/S0040577916060039
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DOI: https://doi.org/10.1134/S0040577916060039