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Dense Subgroups of the Automorphism Groups of Free Algebras

Published online by Cambridge University Press:  20 November 2018

Roger M. Bryant  and
Vesselin Drensky
Roger M. Bryant
Affiliation:
Department of Mathematics University of Manchester Institute of Science and Technology Manchester, M60 1QD United Kingdom
Vesselin Drensky
Affiliation:
Institute of Mathematics Bulgarian Academy of Sciences Sofia Bulgaria
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Abstract

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Let F be the free metabelian Lie algebra of finite rank m over a field K of characteristic 0. The automorphism group Aut F is considered with respect to a topology called the formal power series topology and it is shown that the group of tame automorphisms (automorphisms induced from the free Lie algebra of rank m) is dense in Aut F for m ≥ 4 but not dense for m = 2 and m = 3. At a more general level, we study the formal power series topology on the semigroup of all endomorphisms of an arbitrary (associative or non-associative) relatively free algebra of finite rank m and investigate certain associated modules of the general linear group GLm(AT).

Keywords

17B0117B3017B4017A5017A36
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1135 - 1154
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Anick, D.J., Limits of tame automorphisms ofk[x1,… ,xN], J. Algebra 82(1983), 459–468.Google Scholar
2. Bachmuth, S., Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118(1965), 93–104.Google Scholar
3. Bachmuth, S. and Mochizuki, H.Y, The nonfinite generation ofAut(G), G free metabelian of rank 3, Trans. Amer. Math. Soc. 270(1982), 693–700.Google Scholar
4. Bachmuth, S. and Mochizuki, H.Y, Aut(F) →Aut (F/F“) is surjective for free group F of rank ≥ 4, Trans. Amer. Math. Soc. 292(1985),81-101.Google Scholar
5. Bahturin, Yu. A., Identical relations in Lie algebras, Nauka, Moscow, 1985. Russian; Translation, VNU Science Press, Utrecht, 1987.Google Scholar
6. Bahturin, Yu. A. and Nabiyev, S., Automorphisms and derivations ofabelian extensions of some Lie algebras, Abh. Math. Sem. Univ. Hamburg, 62(1992), 43–57.Google Scholar
7. Chein, O., IA automorphisms of free and free metabelian groups, Comm. Pure Appl. Math. 21(1968), 605–629.Google Scholar
8. Cohn, P.M., Subalgebrasoffree associative algebras, Proc. London Math. Soc. (3) 14(1964), 618–632.Google Scholar
9. Drensky, V. and Gupta, C.K., Automorphisms of free nilpotent Lie algebras, Canad. J. Math. 42(1990), 259–279.Google Scholar
10. Green, J.A., Polynomial representations of GLn, Lecture Notes in Math. 830, Springer-Verlag, Berlin- Heidelberg-New York, 1980.Google Scholar
11. Macdonald, I.G., Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 1979.Google Scholar
12. Roman, V.A.'kov, Groups of automorphisms of free metabelian groups. In: Problems of relations between abstract and applied algebra, Russian, Novosibirsk, (1985), 53–80.Google Scholar
13. Serre, J.-P, Lie algebras and Lie groups, W.A. Benjamin Inc., New York-Amsterdam, 1965.Google Scholar
14. Shmel'kin, A.L., Wreath products of Lie algebras and their application in the theory of groups, Trudy Moskov. Mat. Obshch. 29(1973), 247–260. Russian; Translation, Trans. Moscow Math. Soc. 29(1973), 239–252.Google Scholar