Bulletin of the
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BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(2): 455-461

Online first article March 9, 2017      Printed March 31, 2017

https://doi.org/10.4134/BKMS.b160012

Copyright © The Korean Mathematical Society.

On the structure of factor Lie algebras

Homayoon Arabyani, Farhad Panbehkar, and Hesam Safa

Neyshabur Branch, Islamic Azad University, Neyshabur Branch, Islamic Azad University, University of Bojnord

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 $ \mathrm{dim}L/Z(L)=n$, then $ \mathrm{dim}L_{(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 $s\geq0$) 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 dim$L_{(n+1)}$, when the factor Lie algebra $L/Z_n(L)$ is finitely generated and $n\geq 1$.

Keywords: factor Lie algebra, $n$-isoclinism, nilpotent Lie algebra

MSC numbers: Primary 17B30, 17B60; Secondary 17B99

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