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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the rank of an element of a free Lie algebra

Author: Vladimir Shpilrain
Journal: Proc. Amer. Math. Soc. 123 (1995), 1303-1307
MSC: Primary 17B01; Secondary 17B40
MathSciNet review: 1231044
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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.

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Article copyright: © Copyright 1995 American Mathematical Society