On some properties of the upper central series in Leibniz algebras

Leonid A. Kurdachenko; Javier Otal; Igor Ya. Subbotin

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 2, page 161-175
  • ISSN: 0010-2628

Abstract

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This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra L includes a finite dimensional ideal K such that the factor-algebra L / K is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.

How to cite

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Kurdachenko, Leonid A., Otal, Javier, and Subbotin, Igor Ya.. "On some properties of the upper central series in Leibniz algebras." Commentationes Mathematicae Universitatis Carolinae 60.2 (2019): 161-175. <http://eudml.org/doc/294409>.

@article{Kurdachenko2019,
abstract = {This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra $L$ includes a finite dimensional ideal $K$ such that the factor-algebra $L/K$ is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.},
author = {Kurdachenko, Leonid A., Otal, Javier, Subbotin, Igor Ya.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Leibniz algebra; Lie algebra; center; central serie; hypercenter; nilpotent residual},
language = {eng},
number = {2},
pages = {161-175},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On some properties of the upper central series in Leibniz algebras},
url = {http://eudml.org/doc/294409},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Kurdachenko, Leonid A.
AU - Otal, Javier
AU - Subbotin, Igor Ya.
TI - On some properties of the upper central series in Leibniz algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 2
SP - 161
EP - 175
AB - This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra $L$ includes a finite dimensional ideal $K$ such that the factor-algebra $L/K$ is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.
LA - eng
KW - Leibniz algebra; Lie algebra; center; central serie; hypercenter; nilpotent residual
UR - http://eudml.org/doc/294409
ER -

References

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  1. Baer R., 10.1007/BF01343558, Math. Ann. 124 (1952), 161–177 (German). MR0045720DOI10.1007/BF01343558
  2. Ballester-Bolinches A., Camp-Mora S., Kurdachenko L. A., Otal J., 10.1016/j.jalgebra.2013.06.035, J. Algebra 393 (2013), 1–15. Zbl1294.20045MR3090052DOI10.1016/j.jalgebra.2013.06.035
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  4. Dixon M. R., Kurdachenko L. A., Otal J., 10.1016/j.jalgebra.2015.06.005, J. Algebra 440 (2015), 489–503. MR3373402DOI10.1016/j.jalgebra.2015.06.005
  5. Dixon M. R., Kurdachenko L. A., Otal J., 10.1080/00927872.2016.1175593, Comm. Algebra 45 (2017), no. 1, 234–246. MR3556568DOI10.1080/00927872.2016.1175593
  6. de Falco M., de Giovanni F., Musella C., Sysak Y. P., 10.1090/S0002-9939-2010-10625-1, Proc. Amer. Math. Soc. 139 (2011), no. 2, 385–389. MR2736322DOI10.1090/S0002-9939-2010-10625-1
  7. Franciosi S., de Giovanni F., Kurdachenko L. A., 10.1006/jabr.1995.1155, J. Algebra 174 (1995), no. 3, 823–847. MR1337173DOI10.1006/jabr.1995.1155
  8. Kurdachenko L. A., Otal J., Pypka A. A., 10.1007/s40879-016-0093-5, Eur. J. Math. 2 (2016), no. 2, 565–577. MR3499000DOI10.1007/s40879-016-0093-5
  9. Kurdachenko L. A., Otal J., Subbotin I. Ya., On a generalization of Baer theorem, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2597–2602. MR3056549
  10. Kurdachenko L. A., Pypka A. A., Subbotin I. Ya., On some relations between the factors of the upper and lower central series in Lie algebras, Serdica Math. J. 41 (2015), no. 2–3, 293–306. MR3363607
  11. Kurdachenko L. A., Shumyatsky P., 10.1017/S0305004112000412, Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 63–69. MR3002583DOI10.1017/S0305004112000412
  12. Kurdachenko L. A., Subbotin I. Ya., On the relationships between the factors of upper and lower central series in groups and other algebraic structures, Note Mat. 36 (2016), suppl. 1, 35–50. MR3601080
  13. Neumann B. H., Groups with finite classes of conjugate elements, Proc. London Math. Soc. (3) 1 (1951), 178–187. Zbl0043.02401MR0043779

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