Matrix Representation for Decomposable Solvable Six-Dimensional Lie Algebra

Section: Article
Issue
Vol. 35 No. 1 (2026): Volume 35, Issue 1
Published
Jan 1, 2026
Pages
1-7

Abstract

This paper extends the classification of three-dimensional connected topological proper loops , for which the multiplication group  is a six-dimensional decomposable solvable Lie group. Building upon the significant results presented in [1], which established class two central nilpotency for these loops, we derive explicit matrix representations for the associated Lie algebras. These representations are critical for completing the classification, as they facilitate verification of structural compatibility with the conditions dictated by the Lie bracket. We identify  distinct families of matrix Lie groups, including  groups characterized by one-dimensional centers and  groups with two-dimensional centers. Additionally, we clarify the correspondence between these groups and their derived Lie algebras. Our findings address an existing gap in the literature by providing a systematic framework for examining the interrelationships between topological loops and their corresponding Lie groups.

References

  1. A. Al-Abayechi and . Á. Figula, "Topological loops with Six-dimensional decomposable solvable multiplication groups," vol. 77, no. 20, 2022, doi.org/10.1007/s00025-021-01546-8.
  2. A. A. Albert, "Quasigroups I," Trans. Amer. Math. Soc., no. 54, pp. 507-519, 1943, doi.org/10.1090/S0002-9947-1943-0009962-7.
  3. R. H. Bruck, "Contributions to the Theory of Loops," Trans. Amer. Math. Soc., no. 60, pp. 245-354, 1946, doi.org/10.1090/S0002-9947-1946-0017288-3 .
  4. M. Mazur, "Connected transversals to nilpotent groups," Walter de Gruyter, 2007, DOI 10.1515/JGT.2007.015 .
  5. D. Stanovský and P. Vojtěchovský, "Abelian extensions and solvable loops," Results in Mathematics, vol. 66, no. 3, pp. 367-384, 2014, doi.org/10.1007/s00025-014-0382-6.
  6. M. Niemenmaa, "Finite loops with nilpotent inner mapping groups are centrally nilpotent," Bulletin of the Australian Mathematical Society, vol. 79, no. 1, pp. 109-114, 2009, doi:10.1017/S0004972708001093 .
  7. G. Thompson and A. Shabanskaya, "Solvable extensions of a special class of Nilpotent Lie Algebras," Archivum Mathematicum, no. 49, p. 141–159, 2013, DOI: 10.5817/AM2013-3-141 .
  8. A. Vesanen, "Solvable groups and loops," Journal of Algebra, vol. 180, no. 3, pp. 862-876, 1996, doi.org/10.1006/jabr.1996.0098.
  9. C. Wright, "On the multiplication group of a loop," Illinois Journal of Mathematics, vol. 13, no. 4, pp. 660-673, 1969, DOI: 10.1215/ijm/1256053425.
  10. M. Niemenmaa and K. Tomáš , "On multiplication groups of loops," Journal of Algebra, vol. 135, no. 1, pp. 112-122, 1990, doi.org/10.1016/0021-8693(90)90152-E .
  11. P. Csörgő, "On connected transversals to abelian subgroups and loop theoretical consequences," Archiv der Mathematik, vol. 86, pp. 499-516, 2006, doi.org/10.1007/s00013-006-1379-5 .
  12. A. Drápal, "Orbits of inner mapping groups," Monatshefte für Mathematik, vol. 134, pp. 191-206, 2002, doi.org/10.1007/s605-002-8256-2 .
  13. G. P. Nagy and P. Vojtěchovský, "Moufang loops with commuting inner mappings," Journal of Pure and Applied Algebra, vol. 213, no. 11, pp. 2177-2186, 2009, doi.org/10.1016/j.jpaa.2009.04.004 .
  14. M. Niemenmaa, "On finite loops whose inner mapping groups are abelian," Bulletin of the Australian Mathematical Society, vol. 65, no. 3, pp. 477-484, 2002, doi.org/10.1017/S0004972700020529 .
  15. A. Vesanen, "Finite classical groups and multiplication groups of loops," Mathematical Proceedings of the Cambridge Philosophical Society, vol. 117, no. 3, pp. 425-429, 1995, doi.org/10.1017/S0305004100073278.
  16. P. Nagy and K. Strambach, Loops in group theory and Lie theory, Walter de Gruyter, 2011.
  17. A. Figula, "The multiplication groups of 2-dimensional topological loops," Walter de Gruyter GmbH & Co. KG, pp. 419-429, 2009, doi.org/10.1515/JGT.2008.087 .
  18. A. Figula, "On the Multiplication Groups of Three-Dimensional Topological Loops," J. Lie Theory, no. 21, pp. 385-415, 2011, https://zbmath.org/1222.22020.
  19. A. Figula, "Three-dimensional topological loops with solvable multiplication groups," Communications in Algebra, vol. 42, no. 1, pp. 444-468, 2014, doi.org/10.1080/00927872.2012.717320 .
  20. A. Figula and M. Lattuca, "Three-dimensional topological loops with nilpotent multiplication groups," J. Lie Theory, vol. 25, pp. 787-805, 2015, https://zbmath.org/1331.22007 .
  21. A. Figula and A. Al-Abayechi, "Topological loops with solvable multiplication groups of dimension at most six are centrally nilpotent," International Journal of Group Theory, vol. 9, no. 2, pp. 81-94, 2020, doi.org/10.22108/IJGT.2019.114770.1522.
  22. Á. Figula and . A. Al-Abayechi, "Topological loops having solvable indecomposable Lie groups as their multiplication groups," Transformation Groups, vol. 26, no. 1, pp. 279-303, 2020, doi.org/10.1007/s00031-020-09604-1 .
  23. Á. Figula, K. Ficzere and A. Al-Abayechi, "Topological loops with six-dimensional solvable multiplication groups having five-dimensional nilradical," Annales Mathematicae et Informaticae, vol. 50, pp. 71-87, 2019, doi.org/10.33039/ami.2019.08.001 .
  24. R. Ghanam, . I. Strugar and G. Thompson, "Matrix representation for low dimensional Lie algebras," EXTRACTA MATHEMATICE, vol. 20, no. 2, p. 151–184, 2005.
  25. G. M. Mubarakzyanov, "On Solvable Lie Algebras," Izv. Vyssh. Uchebn. Zaved. Mat. , no. 1, p. 114–123, 1963.
  26. G. M. Mubarakzyanov, " Classification of real structures of Lie Algebras of fifth order," Izv. Vyssh. Uchebn. Zaved. Mat., no. 3, pp. 99-106, 1963.
  27. R. H. Bruck, "Contributions to the theory of loops," Transactions of the American Mathematical Society, vol. 60, no. 2, pp. 245-354, 1946, doi.org/10.1090/S0002-9947-1946-0017288-3.

Author

Ameer Al-Abayechi

Department of Mathematics, Computer Science and Math., University of Kufa, Najaf, Iraq

Identifiers

https://doi.org/10.33899/jes.v35i1.49258
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[1]
A. Al-Abayechi, “Matrix Representation for Decomposable Solvable Six-Dimensional Lie Algebra”, JES, vol. 35, no. 1, pp. 1–7, Jan. 2026.
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