| Title:
|
On some properties of the upper central series in Leibniz algebras (English) |
| Author:
|
Kurdachenko, Leonid A. |
| Author:
|
Otal, Javier |
| Author:
|
Subbotin, Igor Ya. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
161-175 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Leibniz algebra |
| Keyword:
|
Lie algebra |
| Keyword:
|
center |
| Keyword:
|
central serie |
| Keyword:
|
hypercenter |
| Keyword:
|
nilpotent residual |
| MSC:
|
17A32 |
| MSC:
|
17A60 |
| MSC:
|
17A99 |
| idZBL:
|
Zbl 07144885 |
| idMR:
|
MR3982464 |
| DOI:
|
10.14712/1213-7243.2019.009 |
| . |
| Date available:
|
2019-08-05T09:42:24Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147813 |
| . |
| Reference:
|
[1] Baer R.: Endlichkeitskriterien für Kommutatorgruppen.Math. Ann. 124 (1952), 161–177 (German). MR 0045720, 10.1007/BF01343558 |
| Reference:
|
[2] Ballester-Bolinches A., Camp-Mora S., Kurdachenko L. A., Otal J.: Extension of a Schur theorem to groups with a central factor with a bounded section rank.J. Algebra 393 (2013), 1–15. Zbl 1294.20045, MR 3090052, 10.1016/j.jalgebra.2013.06.035 |
| Reference:
|
[3] Chupordia V. A., Kurdachenko L. A., Subbotin I. Ya.: On some “minimal" Leibniz algebras.J. Algebra Appl. 16 (2017), no. 5, 1750082, 16 pages. MR 3634087, 10.1142/S0219498817500827 |
| Reference:
|
[4] Dixon M. R., Kurdachenko L. A., Otal J.: On groups whose factor-group modulo the hypercentre has finite section $p$-rank.J. Algebra 440 (2015), 489–503. MR 3373402, 10.1016/j.jalgebra.2015.06.005 |
| Reference:
|
[5] Dixon M. R., Kurdachenko L. A., Otal J.: On the structure of some infinite dimensional linear groups.Comm. Algebra 45 (2017), no. 1, 234–246. MR 3556568, 10.1080/00927872.2016.1175593 |
| Reference:
|
[6] de Falco M., de Giovanni F., Musella C., Sysak Y. P.: On the upper central series of infinite groups.Proc. Amer. Math. Soc. 139 (2011), no. 2, 385–389. MR 2736322, 10.1090/S0002-9939-2010-10625-1 |
| Reference:
|
[7] Franciosi S., de Giovanni F., Kurdachenko L. A.: The Schur property and groups with uniform conjugacy classes.J. Algebra 174 (1995), no. 3, 823–847. MR 1337173, 10.1006/jabr.1995.1155 |
| Reference:
|
[8] Kurdachenko L. A., Otal J., Pypka A. A.: Relationships between factors of canonical central series of Leibniz algebras.Eur. J. Math. 2 (2016), no. 2, 565–577. MR 3499000, 10.1007/s40879-016-0093-5 |
| Reference:
|
[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. MR 3056549 |
| Reference:
|
[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. MR 3363607 |
| Reference:
|
[11] Kurdachenko L. A., Shumyatsky P.: The ranks of central factor and commutator groups.Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 63–69. MR 3002583, 10.1017/S0305004112000412 |
| Reference:
|
[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. MR 3601080 |
| Reference:
|
[13] Neumann B. H.: Groups with finite classes of conjugate elements.Proc. London Math. Soc. (3) 1 (1951), 178–187. Zbl 0043.02401, MR 0043779 |
| . |
