Skip to main content
SpringerLink
Log in

On the Automorphisms of a Free Lie Algebra of Rank 3 Over an Integral Domain

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We prove that the group of tame automorphisms of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary Euclidean ring analogous to the Anick automorphism [1] of free associative algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Umirbaev U. U., “The Anick automorphism of free associative algebras,” J. Reine Angew. Math., vol. 605, 165–178 (2007).

    MathSciNet  MATH  Google Scholar 

  2. Jung H. W. E., “Über ganze birationale Transformationen der Ebene,” J. Reine Angew. Math., vol. 184, 161–174 (1942).

    MathSciNet  MATH  Google Scholar 

  3. Van der Kulk W., “On polynomial rings in two variables,” Nieuw Arch. Wiskd., vol. 1, no. 3, 33–41 (1953).

    MathSciNet  MATH  Google Scholar 

  4. Shafarevich I. R., “On some infinite-dimensional groups,” Rend. Mat. Appl., vol. 25, 208–212 (1966).

    MathSciNet  Google Scholar 

  5. Wright D., “The amalgamated free product structure of GL2(k[x1,…,xn]),” J. Pure Appl. Algebra, vol. 12, 235–251 (1978).

    Article  MathSciNet  Google Scholar 

  6. Makar-Limanov L. G., “Automorphisms of a free algebra with two generators,” Funct. Anal. Appl., vol. 4, no. 3, 262–264 (1970).

    Article  MathSciNet  Google Scholar 

  7. Czerniakiewicz A. G., “Automorphisms of a free associative algebra of rank 2. I, II,” Trans. Amer. Math. Soc., I: vol. 160, 393–401 (1971); II: vol. 171, 309–315 (1972).

    MathSciNet  MATH  Google Scholar 

  8. Makar-Limanov L., Turusbekova U., and Umirbaev U., “Automorphisms and derivations of free Poisson algebras in two variables,” J. Algebra, vol. 322, no. 9, 3318–3330 (2009).

    Article  MathSciNet  Google Scholar 

  9. Kozybaev D., Makar-Limanov L., and Umirbaev U., “The Freiheitssatz and the automorphisms of free right-symmetric algebras,” Asian-Eur. J. Math., vol. 1, no. 2, 243–254 (2008).

    Article  MathSciNet  Google Scholar 

  10. Alimbaev A. A., Naurazbekova A. S., and Kozybaev D. Kh., “Linearization of automorphisms and triangulation of derivations of free algebras of rank 2,” Sib. Èlektron. Mat. Izv., vol. 16, 1133–1146 (2019).

    Article  MathSciNet  Google Scholar 

  11. Shestakov I. P. and Umirbaev U. U., “Tame and wild automorphisms of rings of polynomials in three variables,” J. Amer. Math. Soc., vol. 17, no. 1, 197–227 (2004).

    Article  MathSciNet  Google Scholar 

  12. Cohn P. M., “Subalgebras of free associative algebras,” Proc. Lond. Math. Soc., vol. 14, no. 3, 618–632 (1964).

    Article  MathSciNet  Google Scholar 

  13. Lewin J., “On Schreier varieties of linear algebras,” Trans. Amer. Math. Soc., vol. 132, 553–562 (1968).

    Article  MathSciNet  Google Scholar 

  14. Kurosh A. G., “Nonassociative free algebras and free products of algebras,” Mat. Sb., vol. 20, no. 2, 239–262 (1947).

    MathSciNet  MATH  Google Scholar 

  15. Shirshov A. I., “Subalgebras of free commutative and free anticommutative algebras,” Mat. Sb., vol. 34, no. 1, 81–88 (1954).

    MathSciNet  MATH  Google Scholar 

  16. Shirshov A. I., “Subalgebras of free Lie algebras,” Mat. Sb., vol. 33, no. 2, 441–452 (1953).

    MathSciNet  MATH  Google Scholar 

  17. Witt E., “Die Unterringe der freien Lieschen Ringe,” Math. Z., Bd 64, 195–216 (1956).

    Article  MathSciNet  Google Scholar 

  18. Mikhalev A. A., “Subalgebras of free colored Lie superalgebras,” Math. Notes, vol. 37, no. 5, 356–360 (1985).

    Article  MathSciNet  Google Scholar 

  19. Shtern A. S., “Free Lie superalgebras,” Sib. Math. J., vol. 27, no. 1, 136–140 (1986).

    Article  MathSciNet  Google Scholar 

  20. Kryazhovskikh G. V. and Kukin G. P., “On subrings of free rings,” Sib. Math. J., vol. 30, no. 6, 903–914 (1989).

    Article  MathSciNet  Google Scholar 

  21. Kryazhovskikh G. V. and Kukin G. P., “Algorithmic properties of free rings,” Sib. Math. J., vol. 32, no. 6, 971–981 (1991).

    Article  MathSciNet  Google Scholar 

  22. Alimbaev A. A. and Umirbaev U. U., “The Nagata automorphism of free nonassociative algebras of rank two over Euclidean domains,” Sib. Èlektron. Mat. Izv., vol. 14, 1279–1288 (2017).

    MathSciNet  MATH  Google Scholar 

  23. Nagata M., On the Automorphism Group of k[x, y], Kyoto Univ., Kinokuniya, Tokyo (1972) (Lect. Math.).

    MATH  Google Scholar 

  24. Cohn P. M., Free Ideal Rings and Localization in General Rings, Cambridge Univ. Press, Cambridge (2006) (New Math. Monogr.; Vol. 3).

    Book  Google Scholar 

  25. Shevelin M. A., “Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions,” Sib. Math. J., vol. 53, no. 5, 934–942 (2012).

    Article  MathSciNet  Google Scholar 

  26. Nauryzbaev R. Zh. and Umirbaev U. U., “Structure of the amalgamated product in automorphism groups of the free Lie algebras of rank 3,” in: Report on the International Conference “Mal’tsev Meeting,” 2018, 158–158. http://www.math.nsc.ru/conference/malmeet/18/maltsev18.pdf.

  27. Zhevlakov K. A., Slinko A. M., Shestakov I. P., and Shirshov A. I., Rings That Are Nearly Associative, Academic Press, New York (1982).

    Google Scholar 

  28. Shirshov A. I., Selected Works. Rings and Algebras [Russian], Nauka, Moscow (1984).

    MATH  Google Scholar 

  29. Magnus W., Karrass A., and Solitar D., Combinatorial Group Theory, Dover Publications, Mineola (2004).

    MATH  Google Scholar 

  30. Vinberg E. B., A Course in Algebra [Russian], Faktorial Press, Moscow (2001).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. U. Sultangazin Kostanai State Pedagogical University, Kostanai, Kazakhstan

    A. A. Alimbaev

  2. L. N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan

    R. Zh. Nauryzbaev

  3. Wayne State University, Detroit, USA

    U. U. Umirbaev

Authors
  1. A. A. Alimbaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Zh. Nauryzbaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. U. U. Umirbaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Alimbaev.

Additional information

Russian Text © The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 1, pp. 3–16.

The authors were partially supported under Project AP05133009 by the Institute of Mathematics and Mathematical Modeling of the Ministry of Education and Science of the Republic of Kazakhstan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alimbaev, A.A., Nauryzbaev, R.Z. & Umirbaev, U.U. On the Automorphisms of a Free Lie Algebra of Rank 3 Over an Integral Domain. Sib Math J 61, 1–10 (2020). https://doi.org/10.1134/S0037446620010012

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446620010012

Keywords

Search

Navigation