Geodesic
Integrable extensions of the Adler map via Grassmann algebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 2, pp. 179-187 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study certain extensions of the Adler map on Grassmann algebras $\Gamma(n)$ of order $n$. We consider a known Grassmann-extended Adler map and under the assumption that $n=1$, obtain a commutative extension of the Adler map in six dimensions. We show that the map satisfies the Yang–Baxter equation, admits three invariants, and is Liouville integrable. We solve the map explicitly by regarding it as a discrete dynamical system.
Keywords: Yang–Baxter map, Grassmann algebra, Liouville integrability, solution of discrete dynamical system, symplectic structure. 
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P. Adamopoulou; S. Konstantinou-Rizos; G. Papamikos. Integrable extensions of the Adler map via Grassmann algebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 2, pp. 179-187. http://geodesic.mathdoc.fr/item/TMF_2021_207_2_a0/

[1] V. M. Bukhshtaber, “Otobrazheniya Yanga–Bakstera”, UMN, 53:6(324) (1998), 241–242 | DOI | DOI | MR | Zbl

[2] E. K. Sklyanin, “O klassicheskikh predelakh $SU(2)$-invariantnykh reshenii uravneniya Yanga–Bakstera”, Zap. nauchn. sem. LOMI, 146 (1985), 119–136 | DOI | MR | Zbl

[3] V. G. Drinfeld, “On some unsolved problems in quantum group theory”, Quantum Groups (Euler International Mathematical Institute, Leningrad, 1990), Lecture Notes in Mathematics, 1510, ed. P. P. Kulish, Springer, Berlin, 2006, 1–8 | DOI | MR

[4] T. E. Kouloukas, V. G. Papageorgiou, “Entwining Yang–Baxter maps and integrable lattices”, Algebra, Geometry and Mathematical Physics (Bȩdlewo, Poland, October 12–16, 2009), Banach Center Publications, 93, eds. V. Abramov, J. Fuchs, E. Paal, A. Stolin, A. Tralle, P. Urbanski, Polish Academy of Sciences, Institute of Mathematics, Warszawa, 2011, 163–175 | DOI | MR | Zbl

[5] T. E. Kouloukas, V. G. Papageorgiou, “Yang–Baxter maps with first-degree-polynomial $2\times 2$ Lax matrices”, J. Phys. A: Math. Theor., 42:40 (2009), 404012, 12 pp. | DOI | MR

[6] Yu. B. Suris, A. P. Veselov, “Lax matrices for Yang–Baxter maps”, J. Nonlinear Math. Phys., 10:suppl. 2 (2003), 223–230 | DOI | MR

[7] V. E. Adler, A. I. Bobenko, Yu. B. Suris, “Geometry of Yang–Baxter maps: pencils of conics and quadrirational mappings”, Commun. Anal. Geom., 12:5 (2004), 967–1007 | DOI | MR

[8] V. G. Papageorgiou, A. G. Tongas, A. P. Veselov, “Yang–Baxter maps and symmetries of integrable equations on quad–graphs”, J. Math. Phys., 47:8 (2006), 083502, 16 pp. | DOI | MR

[9] S. Konstantinou-Rizos, A. V. Mikhailov, “Darboux transformations, finite reduction groups and related Yang–Baxter maps”, J. Phys. A: Math. Theor., 46:42 (2013), 425201, 16 pp. | DOI | MR

[10] S. Konstantinou-Rizos, G. Papamikos, “Entwining Yang–Baxter maps related to NLS type equations”, J. Phys. A: Math. Theor., 52:48 (2019), 485201, 16 pp. | DOI | MR

[11] A. V. Mikhailov, G. Papamikos, J. P. Wang, “Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere”, Lett. Math. Phys., 106:7 (2016), 973–996 | DOI | MR

[12] V. Caudrelier, N. Crampé, Q. C. Zhang, “Set-theoretical reflection equation: classification of reflection maps”, J. Phys. A: Math. Theor., 46:9 (2013), 095203, 12 pp., arXiv: 1210.5107 | DOI | MR

[13] V. Caudrelier, Q. C. Zhang, “Yang–Baxter and reflection maps from vector solitons with a boundary”, Nonlinearity, 27:6 (2014), 1081–1103 | DOI | MR

[14] A. Doikou, A. Smoktunowicz, From braces to Hecke algebras quantum groups, arXiv: 1912.03091

[15] W. Rump, “Modules over braces”, Algebra Discrete Math., 2 (2006), 127–137 | MR

[16] G. G. Grahovski, A. V. Mikhailov, “Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras”, Phys. Lett. A, 377:45–48 (2013), 3254–3259, arXiv: 1303.1853 | DOI | MR

[17] L.-L. Xue, D. Levi, Q. P. Liu, “Supersymmetric KdV equation: Darboux transformation and discrete systems”, J. Phys. A: Math. Theor., 46:50 (2013), 502001, 11 pp. | DOI | MR

[18] L.-L. Xue, Q. P. Liu, “Bäcklund–Darboux transformations and discretizations of super KdV equation”, SIGMA, 10 (2014), 045, 10 pp., arXiv: 1312.6976 | DOI | MR

[19] L.-L. Xue, Q. P. Liu, “A supersymmetric AKNS problem and its Darboux–Bäcklund transformations and discrete systems”, Stud. Appl. Math., 135:1 (2015), 35–62 | DOI | MR

[20] G. G. Grahovski, S. Konstantinou-Rizos, A. V. Mikhailov, “Grassmann extensions of Yang–Baxter maps”, J. Phys. A: Math. Theor., 49:14 (2016), 145202, 17 pp. | DOI | MR

[21] S. Konstantinou-Rizos, T. E. Kouloukas, “A noncommutative discrete potential KdV lift”, J. Math. Phys., 59:6 (2018), 063506, 13 pp., arXiv: 1611.08923 | DOI | MR

[22] S. Konstantinou-Rizos, A. V. Mikhailov, “Anticommutative extension of the Adler map”, J. Phys. A: Math. Theor., 49:30 (2016), 30LT03, 7 pp. | DOI | MR

[23] S. Konstantinou-Rizos, “On the $3D$ consistency of a Grassmann extended lattice Boussinesq system”, Nucl. Phys. B, 951 (2020), 114878, 24 pp. | DOI | MR

[24] F. A. Berezin, Vvedenie v superanaliz, Izd-vo MTsNMO, M., 2013 | DOI | MR

[25] L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Algebras and Superalgebras, Academic Press, San Diego, CA, 2000 | MR | Zbl

[26] V. E. Adler, “Perekroika mnogougolnikov”, Funkts. analiz i ego pril., 27:2 (1993), 79–82 | DOI | MR | Zbl

[27] A. P. Fordy, “Periodic cluster mutations and related integrable maps”, J. Phys. A: Math. Theor., 47:47 (2014), 474003, 44 pp. | DOI | MR

[28] S. Maeda, “Completely integrable symplectic mapping”, Proc. Japan Acad. Ser. A, 63:6 (1987), 198–200 | DOI | MR

[29] A. P. Veselov, “Integriruemye otobrazheniya”, UMN, 46:5(281) (1991), 3–45 | DOI | MR | Zbl