# On the dimension of the product $[L_2,L_2,L_1]$‎ in free Lie algebras

Document Type : Ischia Group Theory 2016

Author

Ahi Evran University

Abstract

Let $L$ be a free Lie algebra of rank $r\geq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$‎. ‎By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$‎, ‎we determine the dimension of $[L_2,L_2,L_1]$‎. ‎Moreover‎, ‎by this method‎, ‎we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.

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