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 Mosc. Math. J., 2003, Volume 3, Number 1, Pages 97–103 (Mi mmj78)

Set-theoretical solutions to the Yang–Baxter relation from factorization of matrix polynomials and $\theta$-functions

A. V. Odesskii

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: New set-theoretical solutions to the Yang–Baxter Relation are constructed. These solutions arise from the decompositions “in different order” of matrix polynomials and $\theta$-functions. We also construct a “local action of the symmetric group” in these cases, generalizations of the action of the symmetric group $S_N$ given by the set-theoretical solution.

Key words and phrases: Yang–Baxter relation, set-theoretical solution, local action of the symmetric group, matrix polynomials, matrix $\theta$-functions.

DOI: https://doi.org/10.17323/1609-4514-2003-3-1-97-103

Full text: http://www.ams.org/.../abst3-1-2003.html
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MSC: 81R50
Received: November 2, 2001; in revised form April 8, 2002
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Citation: A. V. Odesskii, “Set-theoretical solutions to the Yang–Baxter relation from factorization of matrix polynomials and $\theta$-functions”, Mosc. Math. J., 3:1 (2003), 97–103

Citation in format AMSBIB
\Bibitem{Ode03} \by A.~V.~Odesskii \paper Set-theoretical solutions to the Yang--Baxter relation from factorization of matrix polynomials and $\theta$-functions \jour Mosc. Math.~J. \yr 2003 \vol 3 \issue 1 \pages 97--103 \mathnet{http://mi.mathnet.ru/mmj78} \crossref{https://doi.org/10.17323/1609-4514-2003-3-1-97-103} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1996805} \zmath{https://zbmath.org/?q=an:1052.81050} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208594100008} 

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• http://mi.mathnet.ru/eng/mmj/v3/i1/p97

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This publication is cited in the following articles:
1. Borodin A., “Isomonodromy transformations of linear systems of difference equations”, Ann. of Math. (2), 160:3 (2004), 1141–1182
2. Adler V.E., Bobenko A.I., Suris Yu.B., “Geometry of Yang–Baxter maps: pencils of conics and quadrirational mappings”, Comm. Anal. Geom., 12:5 (2004), 967–1007
3. Gelfand I., Retakh V., Serconek S., Wilson R., “On a class of algebras associated to directed graphs”, Selecta Math. (N.S.), 11:2 (2005), 281–295
4. Odesskii A.V., Sokolov V.V., “Compatible Lie brackets related to elliptic curve”, J. Math. Phys., 47:1 (2006), 013506, 14 pp.
5. Maldonado C., Mombelli J.M., “On braided groupoids”, J. Algebra, 307:2 (2007), 677–694
6. Retakh V., Serconek Sh., Wilson R.L., “Construction of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials”, Lie Algebras, Vertex Operator Algebras and their Applications, Contemporary Mathematics Series, 442, 2007, 201–219
7. Tsuboi Z., “Quantum Groups, Yang-Baxter Maps and Quasi-Determinants”, Nucl. Phys. B, 926 (2018), 200–238
8. Bazhanov V.V., Sergeev S.M., “Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups”, Nucl. Phys. B, 926 (2018), 509–543