Abstract:
This paper contains a survey of recent developments in the investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras, along with some new results on the images of word maps on algebraic groups defined over special fields: complex, real, pp-adic (or close to such), or finite.
Bibliography: 174 titles.
The research of the first author was financially supported by the Ministry of Science and Higher Education of the Russian Federation, project 1.661.2016/1.4. The research of the second and third authors was supported by ISF grant 1623/16 and the Emmy Noether Research Institute for Mathematics. The paper was written when the second author visited the MPIM (Bonn). The authors thank all these institutions.
Citation:
N. L. Gordeev, B. È. Kunyavskiǐ, E. B. Plotkin, “Geometry of word equations in simple algebraic groups over special fields”, Russian Math. Surveys, 73:5 (2018), 753–796
\Bibitem{GorKunPlo18}
\by N.~L.~Gordeev, B.~\`E.~Kunyavski{\v\i}, E.~B.~Plotkin
\paper Geometry of word equations in simple algebraic groups over special fields
\jour Russian Math. Surveys
\yr 2018
\vol 73
\issue 5
\pages 753--796
\mathnet{http://mi.mathnet.ru/eng/rm9838}
\crossref{https://doi.org/10.1070/RM9838}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3859398}
\zmath{https://zbmath.org/?q=an:1442.20028}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2018RuMaS..73..753G}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000454772500001}
\elib{https://elibrary.ru/item.asp?id=35601273}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060178009}
Linking options:
https://www.mathnet.ru/eng/rm9838
https://doi.org/10.1070/RM9838
https://www.mathnet.ru/eng/rm/v73/i5/p3
This publication is cited in the following 16 articles:
Mai Hoang Bien, Mojtaba Ramezan-Nassab, Nguyen Mac Nam Trung, “Surjectivity of word maps on special linear groups of degree 2”, Journal of Algebra, 650 (2024), 377
Truong Huu Dung, Tran Nam Son, “On Kursov's Theorem for Matrices Over Division Rings”, Linear Algebra and its Applications, 2024
S. Malev, R. Yavich, R. Shayer, “Evaluations of multilinear polynomials on low rank Jordan algebras”, Commun. Algebr., 50:7 (2022), 2840–2845
N. L. Gordeev, “On sequences of word maps of compact topological groups”, J. Math. Sci., 264:1 (2022), 60
S. Dolfi, A. Singh, M. K. Yadav, “P-power Conjugacy Classes in u(n, q) and t(n, q)”, J. Algebra. Appl., 20:07 (2021), 2150121
Alexei Kanel-Belov, Sergey Malev, Louis Rowen, Roman Yavich, “Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov–Kaplansky Conjecture”, SIGMA, 16 (2020), 071, 61 pp.
N. L. Gordeev, “O posledovatelnostyakh verbalnykh otobrazhenii kompaktnykh topologicheskikh grupp”, Voprosy teorii predstavlenii algebr i grupp. 35, Zap. nauchn. sem. POMI, 492, POMI, SPb., 2020, 94–124
Gnutov F., Gordeev N., “Recursive Sequences of Surjective Word Maps For the Algebraic Groups Pgl(2) and Sl2”, Arch. Math., 114:6 (2020), 609–618
Klyachko A.A., Ryabtseva M.A., “The Dimension of Solution Sets to Systems of Equations in Algebraic Groups”, Isr. J. Math., 237:1 (2020), 141–154
R. V. Skuratovskii, “The derived subgroups of Sylow 2-subgroups of the alternating group, commutator width of wreath product of groups”, Mathematics, 8:4 (2020), 472
S. Malev, C. Pines, “The images of multilinear non-associative polynomials evaluated on a rock-paper-scissors algebra with unit over an arbitrary field and its subalgebras”, Chebyshevskii sb., 21:4 (2020), 129–139
E. A. Egorchenkova, “Word Maps of Chevalley Groups Over Infinite Fields”, J Math Sci, 247:4 (2020), 571
F. A. Gnutov, N. L. Gordeev, “On the Image of a Word Map with Constants of a Simple Algebraic Group”, J Math Sci, 247:4 (2020), 550
W. Cocke, M.-C. Ho, “Word maps in finite simple groups”, Arch. Math. (Basel), 113:6 (2019), 565–570
E. A. Egorchenkova, “Verbalnye otobrazheniya grupp Shevalle nad beskonechnymi polyami”, Voprosy teorii predstavlenii algebr i grupp. 34, Zap. nauchn. sem. POMI, 478, POMI, SPb., 2019, 108–127
F. A. Gnutov, N. L. Gordeev, “Ob obraze verbalnogo otobrazheniya s konstantami prostoi algebraicheskoi gruppy”, Voprosy teorii predstavlenii algebr i grupp. 34, Zap. nauchn. sem. POMI, 478, POMI, SPb., 2019, 78–99
Citation in format AMSBIB
Contact us: