On the dimension of the product $[L_۲,L_۲,L_۱]$‎ in free Lie algebras

Let $L$ be a free Lie algebra of rank $r\geq۲$ 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 $[L_۲,L_۲,L_۱]$‎. ‎Moreover‎, ‎by this method‎, ‎we show that the dimension of $[L_۲,L_۲,L_۱]$ over a field of characteristic $۲$ is different from the dimension over a field of characteristic other than $۲$.

کلیدواژه ها:

Free Lie algebra ، homogeneous and fine homogeneous components ، free centre-by-metabelian Lie algebra ، second derived ideal