Almost primitive elements of free Lie algebras of small ranks

A. V. Klimakov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 1, pp. 63-74

Résumé

Let $K$ be a field, $X=\{x_1,\ldots,x_n\}$, and let $L(X)$ be the free Lie algebra over $K$ with the set $X$ of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free, A. I. Shirshov proved that subalgebras of free Lie algebras are free. A subset $M$ of nonzero elements of the free Lie algebra $L(X)$ is said to be primitive if there is a set $Y$ of free generators of $L(X)$, $L(X)=L(Y)$, such that $M\subseteq Y$ (in this case we have $|Y|=|X|=n$). Matrix criteria for a subset of elements of free Lie algebras to be primitive and algorithms to construct complements of primitive subsets of elements with respect to sets of free generators have been constructed. A nonzero element $u$ of the free Lie algebra $L(X)$ is said to be almost primitive if $u$ is not a primitive element of the algebra $L(X)$, but $u$ is a primitive element of any proper subalgebra of $L(X)$ that contains it. A series of almost primitive elements of free Lie algebras has been constructed. In this paper, for free Lie algebras of rank $2$ criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed.