DOI QR코드

DOI QR Code

ON THE STRUCTURE OF FACTOR LIE ALGEBRAS

  • Arabyani, Homayoon (Department of Mathematics Neyshabur Branch Islamic Azad University) ;
  • Panbehkar, Farhad (Department of Mathematics Neyshabur Branch Islamic Azad University) ;
  • Safa, Hesam (Department of Mathematics Faculty of Basic Sciences University of Bojnord)
  • Received : 2016.01.08
  • Published : 2017.03.31

Abstract

The Lie algebra analogue of Schur's result which is proved by Moneyhun in 1994, states that if L is a Lie algebra such that dimL/Z(L) = n, then $dimL_{(2)}={\frac{1}{2}}n(n-1)-s$ for some non-negative integer s. In the present paper, we determine the structure of central factor (for s = 0) and the factor Lie algebra $L/Z_2(L)$ (for all $s{\geq}0$) of a finite dimensional nilpotent Lie algebra L, with n-dimensional central factor. Furthermore, by using the concept of n-isoclinism, we discuss an upper bound for the dimension of $L/Z_n(L)$ in terms of $dimL_{(n+1)}$, when the factor Lie algebra $L/Z_n(L)$ is finitely generated and $n{\geq}1$.

References

  1. J. M. Ancochea-Bermudez and M. Goze, Classification des algebres de Lie nilpotentes de dimension 7, Arch. Math. 52 (1989), no. 2, 157-185.
  2. H. Arabyani and F. Saeedi, On dimensions of derived algebra and central factor of a Lie algebra, Bull. Iranian Math. Soc. 41 (2015), no. 5, 1093-1102.
  3. P. Batten, K. Moneyhum, and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, Comm. Algebra 24 (1996), no. 14, 4319-4330. https://doi.org/10.1080/00927879608825817
  4. Ya. G. Berkovich, On the order of the commutator subgroups and the Schur multiplier of a finite p-group, J. Algebra 144 (1991), no. 2, 269-272. https://doi.org/10.1016/0021-8693(91)90106-I
  5. S. Cicalo, W. A. de Graaf, and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl. 436 (2012), no. 1, 163-189. https://doi.org/10.1016/j.laa.2011.06.037
  6. J. Dixmier, Sur les representations unitaires des groupes de Lie nilpotents III, Canad. J. Math. 10 (1958), 321-348. https://doi.org/10.4153/CJM-1958-033-5
  7. K. Erdmann and M. J. Wildon, Introduction to Lie Algebras, Springer Undergraduate Mathematics Series, 2006.
  8. N. S. Hekster, On the structure of n-isoclinism classes of groups, J. Pure Appl. Algebra 40 (1986), no. 1, 63-85. https://doi.org/10.1016/0022-4049(86)90030-7
  9. I. M. Isaacs, Derived subgroups and centers of capable groups, Proc. Amer. Math. Soc. 129 (2001), no. 10, 2853-2859. https://doi.org/10.1090/S0002-9939-01-05888-9
  10. S. O. Kim, On some finite p-groups, Bull. Korean Math. Soc. 41 (2004), no. 1, 147-151. https://doi.org/10.4134/BKMS.2004.41.1.147
  11. K. Moneyhun, Isoclinism in Lie algebras, Algebras Groups Geom. 11 (1994), no. 1, 9-22.
  12. K. Podoski and B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3441-3445. https://doi.org/10.1090/S0002-9939-05-07663-X
  13. A. R. Salemkar and F. Mirzaei, Characterizing n-isoclinism classes of Lie algebras, Comm. Algebra 38 (2010), no. 9, 3392-3403. https://doi.org/10.1080/00927870903117535
  14. I. Schur, Uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substi-tutionen, J. Reine Angew. Math. 127 (1904), 20-50.
  15. Gr. Tsagas and A. Kobotis, Characteristic elements of a category of nilpotent Lie alge-bras of dimension eight, Algebra Groups Geom. 9 (1992), no. 3, 137-256.
  16. J. Wiegold, Multiplicators and groups with finite central factor groups, Math. Z. 89 (1965), 345-347. https://doi.org/10.1007/BF01112166