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.

Location: http://mech.math.msu.su/~fpm/eng/k13/k131/k13106h.htm

Last modified: September 5, 2013