On the rank of an element of a free Lie algebra
Author:
Vladimir Shpilrain
Journal:
Proc. Amer. Math. Soc. 123 (1995), 13031307
MSC:
Primary 17B01; Secondary 17B40
MathSciNet review:
1231044
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [1]
Yu.
A. Bahturin, Identical relations in Lie algebras, VNU Science
Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR 886063
(88f:17032)
 [2]
Anastasia
J. Czerniakiewicz, Automorphisms of a free associative
algebra of rank 2. II, Trans. Amer. Math.
Soc. 171 (1972),
309–315. MR 0310021
(46 #9124), http://dx.doi.org/10.1090/S00029947197203100212
 [3]
Warren
Dicks, A commutator test for two elements to generate the free
algebra of rank two, Bull. London Math. Soc. 14
(1982), no. 1, 48–51. MR 642424
(83g:16005), http://dx.doi.org/10.1112/blms/14.1.48
 [4]
Ralph
H. Fox, Free differential calculus. I. Derivation in the free group
ring, Ann. of Math. (2) 57 (1953), 547–560. MR 0053938
(14,843d)
 [5]
Christophe
Reutenauer, Applications of a noncommutative Jacobian matrix,
J. Pure Appl. Algebra 77 (1992), no. 2,
169–181. MR 1149019
(93a:16021), http://dx.doi.org/10.1016/00224049(92)90083R
 [6]
Vladimir
Shpilrain, On generators of
𝐿/𝑅² Lie algebras, Proc.
Amer. Math. Soc. 119 (1993), no. 4, 1039–1043. MR 1154249
(94a:17002), http://dx.doi.org/10.1090/S0002993919931154249X
 [1]
 Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, Utrecht, 1987. MR 886063 (88f:17032)
 [2]
 A. J. Czerniakiewicz, Automorphisms of free algebras of rank two, II, Trans. Amer. Math. Soc. 171 (1972), 309315. MR 0310021 (46:9124)
 [3]
 W. Dicks, A commutator test for two elements to generate the free algebra of rank two, Bull. London Math. Soc. 14 (1982), 4851. MR 642424 (83g:16005)
 [4]
 R. H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547560. MR 0053938 (14:843d)
 [5]
 C. Reutenauer, Applications of a noncommutative Jacobian matrix, J. Pure Appl. Algebra 77 (1992), 169181. MR 1149019 (93a:16021)
 [6]
 V. Shpilrain, On generators of Lie algebras, Proc. Amer. Math. Soc. 119 (1993), 10391043. MR 1154249 (94a:17002)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
17B01,
17B40
Retrieve articles in all journals
with MSC:
17B01,
17B40
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512310446
PII:
S 00029939(1995)12310446
Article copyright:
© Copyright 1995
American Mathematical Society
