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 , 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 , to have rank is equivalent to another property which in the most interesting and important case when the algebra has a finite rank looks as follows: an arbitrary endomorphism of is an automorphism if and only if *u* belongs to . 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