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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the rank of an element of a free Lie algebra
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by Vladimir Shpilrain PDF
Proc. Amer. Math. Soc. 123 (1995), 1303-1307 Request permission

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.
References
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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