On the rank of an element of a free Lie algebra

Abstract:

Let L be a free Lie algebra over an arbitrary field K, and let $\{ {x_1}, \ldots ,{x_n}, \ldots \} ,n \geq 2$, be a free basis of L. We define the rank of an element u of L as the least number of free generators on which the image of u under an arbitrary automorphism of L can depend. We prove that for a homogeneous element u of degree $m \geq 2$, to have rank $n \geq 2$ is equivalent to another property which in the most interesting and important case when the algebra $L = {L_n}$ has a finite rank $n \geq 2$ looks as follows: an arbitrary endomorphism $\phi$ of ${L_n}$ is an automorphism if and only if u belongs to ${(\phi ({L_n}))^m}$. This yields in particular a simple algorithm for finding the rank of a homogeneous element and also for finding a particular automorphic image of this element realizing the rank.