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 $n\geq 1$.