Advertisement

Journal of Mathematical Sciences

, Volume 206, Issue 6, pp 660–667 | Cite as

On Tame and Wild Automorphisms of Algebras

  • C. K. Gupta
  • V. M. Levchuk
  • Yu. Yu. Ushakov
Article
  • 41 Downloads

Abstract

Let B n be a polynomial algebra of n variables over a field F. Considering a free associative algebra A n of rank n over F as a polynomial algebra of noncommuting variables, we choose the ideal R of all polynomials with a zero absolute term in B n and A n . The well-known concept of wild automorphisms of the algebras A n and B n is transferred to R; the study of wild automorphisms is reduced to monic automorphisms of the algebra R, i.e., those identical on each factor R k /R k+1. In particular, this enables us to study the properties of the known Nagata and Anik automorphisms in detail. For n = 3 we investigate the hypothesis that the Anik automorphism is tame modulo R k for every given integer k > 1.

Keywords

Tame Chevalley Group Polynomial Algebra Free Associative Algebra Jacobian Conjecture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. A. Artamonov, “Nilpotency. Projectivity. Freedom,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 50–53 (1971).Google Scholar
  2. 2.
    E. I. Bunina, “Automorphisms of Chevalley groups of certain types over local rings,” Usp. Mat. Nauk, 62, No. 5, 143–144 (2007).CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. G. Chernyakiewicz, “Automorphisms of a free associative algebra of rank 2,” Trans. Am. Math. Soc., 160, 393–401 (1971); 171, 309–315 (1972).Google Scholar
  4. 4.
    P. M. Cohn, Free Rings and Their Relations, Academic Press, London (1985).MATHGoogle Scholar
  5. 5.
    P. M. Cohn, “Subalgebras of free associative algebras,” Proc. London Math. Soc., 3, No. 14, 618–632 (1964).CrossRefGoogle Scholar
  6. 6.
    R. Crowell and R. Fox, Knot Theory [Russian translation], Mir, Moscow (1967).Google Scholar
  7. 7.
    W. Dicks and J. Lewin, “A Jacobian conjecture for free associative algebras,” Commun. Algebra, 10, No. 12, 1285–1306 (1982).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    V. Drensky and Ch. K. Gupta, “Automorphisms of free nilpotent Lie algebras,” Can. J. Math., 42, No. 2, 259–279 (1990).CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    R. Dubish and S. Perlis, “On total nilpotent algebras,” Am. J. Math., 73, No. 3, 439–452 (1951).CrossRefGoogle Scholar
  10. 10.
    N. Gupta, Free Group Rings, Contemp. Math., Vol. 66, Amer. Math. Soc., Providence.Google Scholar
  11. 11.
    C. K. Gupta, V. M. Levchuk, and Yu. Yu. Ushakov, “Hypercentral and monic automorphisms of classical algebras, rings and groups,” J. SFU. Phys. & Maths, 4, No. 1, 380–390 (2008).Google Scholar
  12. 12.
    A. J. Hahn, D. G. James, and B. Weisfeiler, “Homomorphisms of algebraic and classical groups: a survey,” Can. Math. Soc. Conf. Proc., 4, 249–296 (1984).MathSciNetGoogle Scholar
  13. 13.
    Kourovka Notebook (Unsolved Problems of Group Theory), [in Russian], NSU, Novosibirsk (2010).Google Scholar
  14. 14.
    V. M. Levchuk, “Automorphisms of unipotent subgroups of Chevalley groups,” Algebra Logika, 29, No. 2, 141–161 (1990); 29, No. 3, 315–338 (1990).Google Scholar
  15. 15.
    L. G. Makar-Limanov, “On the automorphisms of a free algebra with 2 generators,” Funkts. Anal. Prilozh., 4, No. 3, 107–108 (1970).MATHMathSciNetGoogle Scholar
  16. 16.
    Yu. I. Merzlyakov, ed., Automorphisms of Classical Groups [in Russian], Mir, Moscow (1976).Google Scholar
  17. 17.
    M. Nagata, On Automorphism Group of k[x, y], Lect. Math., No. 5, Dept. of Maths of Kyoto Univ., Tokyo (1972).Google Scholar
  18. 18.
    V. A. Romankov, “Theorem on an inverse function for free associative algebras,” Sib. Mat. Zh., 45, No. 5, 1178–1183 (2004).MathSciNetGoogle Scholar
  19. 19.
    A. H. Schofield, Representation of Rings over Skew Fields, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, Cambridge (1985).CrossRefMATHGoogle Scholar
  20. 20.
    I. P. Shestakov and U. U. Umirbaev, “The Nagata automorphism is wild,” Proc. Natl. Acad. Sci. USA, 100, No. 22, 12561–12563 (2003).CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    M. K. Smith, “Stably tame automorphisms,” J. Pure Appl. Algebra, 58, 209–212 (1989).CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    U. U. Umirbaev, “Defining relations of a group of tame automorphisms of the polynomial algebra and wild automorphisms of free associative algebras,” Dokl. Ross. Akad. Nauk, 407, No. 3, 319–324 (2006).MathSciNetGoogle Scholar
  23. 23.
    Yu. Yu. Ushakov, Automorphisms of Free Algebras and Functions on Lie Type Groups of Rank 1, Ph.D. Thesis, Sib. Federal Univ., Krasnoyarsk (2013).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • C. K. Gupta
    • 1
  • V. M. Levchuk
    • 2
  • Yu. Yu. Ushakov
    • 2
  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada
  2. 2.Institute of Mathematics and Computer ScienceSiberian Federal UniversityKrasnoyarskRussia

Personalised recommendations