FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2013, VOLUME 18, NUMBER 1, PAGES 63-74

Almost primitive elements of free Lie algebras of small ranks

A. V. Klimakov

Abstract

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Let $K$ be a field, $X\; =\; \{x$₁,¼,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 $M$Í 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.

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Last modified: September 5, 2013