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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
DOI: https://doi.org/10.1090/S0002-9939-1995-1231044-6
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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DOI: https://doi.org/10.1090/S0002-9939-1995-1231044-6
Article copyright: © Copyright 1995 American Mathematical Society

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