Topological loops with solvable multiplication groups of dimension at most six are centrally nilpotent

Document Type : Ischia Group Theory 2018

Authors

Institute of Mathematics, University of Debrecen, Debrecen, Hungary

Abstract

The main result of our consideration is the proof of the centrally nilpotency of class two property for connected topological proper loops $L$ of dimension $\le 3$ which have an at most six-dimensional solvable indecomposable Lie group as their multiplication group. This theorem is obtained from our previous classification by the investigation of six-dimensional indecomposable solvable multiplication Lie groups having a five-dimensional nilradical. We determine the Lie algebras of these multiplication groups and the subalgebras of the corresponding inner mapping groups.

Keywords

Main Subjects


[1] A. A. Albert, Quasigroups I, Trans. Amer. Math. Soc., 54 (1943) 507–519.
[2] R. H. Bruck, Contributions to the Theory of Loops, Trans. Amer. Math. Soc., 60 (1946) 245–354.
[3] A. Figula, The multiplication groups of 2-dimensional topological loops, ´ J. Group Theory, 12 (2009) 419–429.
[4] A. Figula, Three-dimensional topological loops with solvable multiplication groups, ´ Comm. Algebra, 42 (2014) 444–
468.
[5] A. Figula and M. Lattuca, Three-dimensional topological loops with nilpotent multiplication groups, ´ J. Lie Theory,
25 (2015) 787–805.
[6] A. Figula, Quasi-simple Lie groups as multiplication groups of topological loops, ´ Adv. Geom., 15 (2015) 315–331.
[7] A. A. Figula, Al-Abayechi: Topological loops having solvable indecomposable Lie groups as their multiplication ´
groups, submitted to Transform. Groups, 2018.
[8] K. H. Hofmann and K. Strambach, Topological and analytical loops, In: Quasigroups and Loops: Theory and Applications (Eds. O. Chein, H. O. Pflugfelder and J. D. H. Smith), 205–262, Heldermann-Verlag, Berlin, 1990.
[9] G. M. Mubarakzyanov, Classification of Solvable Lie Algebras in dimension six with one non-nilpotent basis element,
Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1963) 104–116.
[10] P. T. Nagy and K. Strambach, Loops in Group Theory and Lie Theory de Gruyter Expositions in Mathematics, 35,
Walter de Gruyter GmbH & Co. KG, Berlin, 2002.
[11] M. Niemenmaa and T. Kepka, On Multiplication Groups of Loops, J. Algebra, 135 (1990) 112–122.
[12] A. Shabanskaya and G. Thompson, Six-dimensional Lie algebras with a five-dimensional nilradical, J. of Lie Theory,
23 (2013) 313–355.
[13] A. Vesanen, Solvable loops and groups, J. Algebra, 180 (1996) 862–876. 
Volume 9, Issue 2 - Serial Number 2
Proceedings of the Ischia Group Theory 2018- Part 2
June 2020
Pages 81-94
  • Receive Date: 28 December 2018
  • Revise Date: 13 March 2019
  • Accept Date: 28 March 2019
  • Published Online: 01 June 2020