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Mosc. Math. J., 2006, Volume 6, Number 1, Pages 57–84 (Mi mmj235)  

This article is cited in 19 scientific papers (total in 20 papers)

$n$-valued groups: theory and applications

V. M. Buchstaber

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We give a survey of the most important results in the theory of $n$-valued groups and some of their applications. Main directions of advanced research will be discussed.
We start with basic definitions. Further exposition follows a sequence of instructive examples originating from various branches of mathematics: algebra, analysis, representation theory, topology, and dynamical systems.
The contents is accessible to a broad audience.

Key words and phrases: Multivalued multiplication and action, hypergroups, deformation of a group, algebraic representations, generalized shift, integrable multivalued dynamical systems.


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MSC: 20N20, 16S34, 05E30
Received: April 20, 2006

Citation: V. M. Buchstaber, “$n$-valued groups: theory and applications”, Mosc. Math. J., 6:1 (2006), 57–84

Citation in format AMSBIB
\by V.~M.~Buchstaber
\paper $n$-valued groups: theory and applications
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 1
\pages 57--84

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    This publication is cited in the following articles:
    1. V. M. Buchstaber, A. A. Gaifullin, “Representations of $m$-valued groups on triangulations of manifolds”, Russian Math. Surveys, 61:3 (2006), 560–562  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. M. Vershik, “Krein Duality, Positive 2-Algebras, and Dilation of Comultiplications”, Funct. Anal. Appl., 41:2 (2007), 99–114  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. A. Gaifullin, P. V. Yagodovskii, “Integrability of $m$-valued dynamics by means of single-generated $m$-valued groups”, Russian Math. Surveys, 62:1 (2007), 181–183  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. P. V. Yagodovskii, “Dual multivalued groups”, Russian Math. Surveys, 64:5 (2009), 955–957  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Dragović V., “Multivalued hyperelliptic continued fractions of generalized Halphen type”, Int. Math. Res. Not. IMRN, 2009, no. 10, 1891–1932  mathscinet  zmath  isi
    6. Pillay A., “A remark on multivalued algebraic groups”, Proc. Amer. Math. Soc., 137:7 (2009), 2175–2180  crossref  mathscinet  zmath  isi  scopus
    7. P. V. Yagodovsky, “Duality in the theory of finite commutative multivalued groups”, J. Math. Sci. (N. Y.), 174:1 (2011), 97–119  mathnet  crossref
    8. Dragović V., “Geometrization and generalization of the Kowalevski top”, Comm. Math. Phys., 298:1 (2010), 37–64  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. Dragović V., Kukić K., “New examples of systems of the Kowalevski type”, Regul. Chaotic Dyn., 16:5 (2011), 484–495  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms”, Trans. Moscow Math. Soc., 72 (2011), 97–142  mathnet  crossref  mathscinet  zmath  elib
    11. V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms, II”, Trans. Moscow Math. Soc., 73 (2012), 167–182  mathnet  crossref  mathscinet  zmath  elib
    13. Dragovic V., “Algebro-Geometric Approach to the Yang–Baxter Equation and Related Topics”, Publ. Inst. Math.-Beograd, 91:105 (2012), 25–48  crossref  mathscinet  zmath  isi  scopus
    14. A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. Proc. Steklov Inst. Math., 286 (2014), 224–239  mathnet  crossref  crossref  isi  elib  elib
    16. Dragovic V., Kukic K., “Discriminantly Separable Polynomials and Quad-Equations”, J. Geom. Mech., 6:3 (2014), 319–333  crossref  mathscinet  zmath  isi  elib  scopus
    17. Dragovic V., Kukic K., “Role of Discriminantly Separable Polynomials in Integrable Dynamical Systems”, Tim 2013 Physics Conference, AIP Conference Proceedings, 1634, eds. Bunoiu O., Avram N., Popescu A., Amer Inst Physics, 2014, 3–8  crossref  isi
    18. Vladimir Dragović, Milena Radnović, “Caustics of Poncelet Polygons and Classical Extremal Polynomials”, Regul. Chaotic Dyn., 24:1 (2019), 1–35  mathnet  crossref
    19. D. V. Gugnin, “Razvetvlennye nakrytiya mnogoobrazii i $\boldsymbol{nH}$-prostranstva”, Funkts. analiz i ego pril., 53:2 (2019), 68–71  mathnet  crossref  elib
    20. V. M. Buchstaber, A. P. Veselov, “Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups”, Russian Math. Surveys, 74:3 (2019), 387–430  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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