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Generic, almost primitive and test elements of free Lie algebras
Author(s):
Alexander
A.
Mikhalev;
Ualbai
Umirbaev;
Jie-Tai
Yu
Journal:
Proc. Amer. Math. Soc.
130
(2002),
1303-1310.
MSC (2000):
Primary 17B01;
Secondary 17B40
Posted:
December 20, 2001
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Abstract:
We construct a series of generic elements of free Lie algebras. New almost primitive and test elements were found. We present an example of an almost primitive element which is not generic.
References:
-
- 1.
- Yu. A. Bahturin, Identical relations in Lie algebras. VNU Science Press, Utrecht, 1987. MR 88f:17032
- 2.
- L. A. Bokut' and G. P. Kukin, Algorithmic and Combinatorial Algebra. Kluwer Academic Publ., Dordrecht, 1994. MR 95i:17002
- 3.
- A. M. Brunner, R. G. Burns, and S. Oates-Williams, On almost primitive elements of free groups with an application to Fuchsian groups. Can. J. Math. 45 (1993), 225-254. MR 94f:20047
- 4.
- P. M. Cohn, Subalgebras of free associative algebras. Proc. London Math. Soc. (3) 14 (1964), 618-632. MR 29:4777
- 5.
- L. P. Comerford, Generic elements of free groups. Arch. Math. (Basel) 65 (1995), no. 3, 185-195. MR 96h:20049
- 6.
- A. Dold, Nullhomologus words in free groups which are not nullhomologus in any proper subgroup. Arch. Math. 50 (1988), 564-569. MR 89f:20035
- 7.
- B. Fine, G. Rosenberger, D. Spellman, and M. Stille. Test, generic and almost primitive elements in free groups. Mat. Contemp. 14 (1998), 45-59. MR 2000b:20028
- 8.
- B. Fine, G. Rosenberger, D. Spellman, and M. Stille. Test words, generic elements and almost primitivity. Pacif. J. Math. 190 (1999), 277-297. MR 2000j:20035
- 9.
- R. C. Lyndon, The equation
 in free groups. Michigan Math. J. 6 (1959), 89-95. MR 21:1999 - 10.
- A. A. Mikhalev and J.-T. Yu, Test elements and retracts of free Lie algebras. Commun. Algebra 25 (1997), 3283-3289. MR 98i:17004
- 11.
- A. A. Mikhalev and J.-T. Yu, Test elements, retracts and automorphic orbits of free algebras. Internat. J. Algebra Comput. 8 (1998), 295-310. MR 99h:17002
- 12.
- A. A. Mikhalev and J.-T. Yu, Primitive, almost primitive, test, and
 -primitive elements of free algebras with the Nielsen-Schreier property. J. Algebra 228 (2000), 603-623. MR 2001f:17007 - 13.
- A. A. Mikhalev and A. A. Zolotykh, Rank and primitivity of elements of free color Lie (
 -)superalgebras. Intern. J. Algebra Comput. 4 (1994), 617-656. MR 95k:17003 - 14.
- A. A. Mikhalev and A. A. Zolotykh, Test elements for monomorphisms of free Lie algebras and superalgebras. Commun. Algebra 23 (1995), 4995-5001. MR 96g:17002
- 15.
- A. A. Mikhalev and A. A. Zolotykh, Combinatorial Aspects of Lie Superalgebras. CRC Press, Boca Raton, New York, 1995. MR 97j:17006
- 16.
- C. Reutenauer, Free Lie Algebras. Clarendon Press, Oxford, 1993. MR 94j:17002
- 17.
- G. Rosenberger, Alternierende produkte in freien Gruppen. Pacif. J. Math. 78 (1978), 243-250. MR 80d:20026
- 18.
- G. Rosenberger, Über Darstellungen von Elementen und Untergruppen in freien Produkten, Lect. Notes Math. 1098 (1984), 142-160. MR 86m:20030
- 19.
- A. I. Shirshov, Subalgebras of free Lie algebras. Mat. Sb. 33 (1953), 441-452. MR 15:596d
- 20.
- V. Shpilrain, Recognizing automorphisms of the free groups. Arch. Math. (Basel) 62 (1994), 385-392. MR 95f:20061
- 21.
- V. Shpilrain, Test elements for endomorphisms of the free groups and algebras. Israel J. Math. 92 (1995), 307-316. MR 96k:20048
- 22.
- J. Stallings, Problems about free quotients of groups. In Geometric Group Theory, Walter de Gruyter, Berlin, 1995, 165-182. MR 97b:20028
- 23.
- J. Stallings, Generic elements in certain groups. In Groups - Korea'94 (Pusan), Walter de Gruyter, Berlin, 1995, 289-294. MR 98m:20025
- 24.
- E. C. Turner, Test words for automorphisms of free groups. Bull. London Math. Soc. 28 (1996), 255-263. MR 96m:20039
- 25.
- E. Witt, Die Unterringe der freien Lieschen Ringe. Math. Z. 64 (1956), 195-216. MR 17:1050a
- 26.
- H. Zieschang, Alternierende Produkte in freien Gruppen. Abh. math. Sem. Univ. Hamburg 27 (1964), 13-31. MR 28:5105
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Additional Information:
Alexander
A.
Mikhalev
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong -- and -- Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia
Email:
sasha@submaths.hku.hk, aamikh@cnit.math.msu.su
Ualbai
Umirbaev
Affiliation:
Department of Mathematics, South Kazakhstan State University, Tauke khana 5, Shymkent, 486050, Kazakhstan
Jie-Tai
Yu
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Email:
yujt@hkusua.hku.hk
DOI:
10.1090/S0002-9939-01-06203-7
PII:
S 0002-9939(01)06203-7
Received by editor(s):
July 5, 2000
Received by editor(s) in revised form:
November 8, 2000
Posted:
December 20, 2001
Additional Notes:
The first author was partially supported by CRCG Research Grant 25500/301/01, by RFBR and INTAS
The third auhor was partially supported by RGC Research Grant HKU 7134/00P
Communicated by:
Lance W. Small
Copyright of article:
Copyright
2001,
American Mathematical Society
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