Skip to main content
SpringerLink
Log in

On invariants of modular free Lie algebras

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Suppose that L(X) is a free Lie algebra of finite rank over a field of positive characteristic. Let G be a nontrivial finite group of homogeneous automorphisms of L(X). It is known that the subalgebra of invariants H = L G is infinitely generated. Our goal is to describe how big its free generating set is. Let \( Y = \bigcup\limits_{n = 1}^\infty {{Y_n}} \) be a homogeneous free generating set of H, where elements of Y n are of degree n with respect to X. We describe the growth of the generating function of Y and prove that |Y n | grow exponentially.

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. Yu. A. Bahturin, Identical Relations in Lie Algebras, VNV Science Press, Utrecht, 1987.

    MATH  Google Scholar 

  2. R. M. Bryant, “On the fixed points of a finite group acting on a free Lie algebra,” J. London Math. Soc. (2), 43, No. 2, 215–224 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  3. R. M. Bryant, “Free Lie algebras and formal power series,” J. Algebra, 253, No. 1, 167–188 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  4. R. M. Bryant and A. I. Papistas, “On the fixed points of a finite group acting on a relatively free Lie algebra,” Glasgow Math. J., 42, 167–181 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  5. V. Drensky, “Fixed algebras of residually nilpotent Lie algebras,” Proc. Am. Math. Soc., 120, No. 4, 1021–1028 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. I. Markushevich and L. A. Markushevich, Introduction to the Theory of Analytic Functions [in Russian], Prosveshchenie, Moscow (1977).

    MATH  Google Scholar 

  7. A. A. Mikhalev, “On fixed points of a free chromatic Lie superalgebra under the action of a finite group of linear automorphisms,” Russ. Math. Surv., 47, No. 4, 220–221 (1992).

    Article  MathSciNet  Google Scholar 

  8. V. M. Petrogradsky, “Growth of finitely generated polynilpotent Lie algebras and groups, generalized partitions, and functions analytic in the unit circle,” Int. J. Algebra Comput., 9, No. 2, 179–212 (1999).

    Article  MathSciNet  Google Scholar 

  9. V. M. Petrogradsky, “On invariants of actions of finite groups on free Lie algebras,” Sib. Math. J., 41, No. 4, 763–770 (2000).

    Article  Google Scholar 

  10. V. M. Petrogradsky, “On Witt’s formula and invariants for free Lie superalgebras,” in: D. Krob, ed., Formal Power Series and Algebraic Combinatorics, Proc. of the 12th Int. Conf. FPSAC ’00, Moscow, Russia, June 26–30, 2000, Springer, Berlin (2000), pp. 543–551.

    Google Scholar 

  11. V. M. Petrogradsky, “Characters and invariants for free Lie superalgebras,” St. Petersburg Math. J., 13, No. 1, 107–122 (2002).

    MATH  MathSciNet  Google Scholar 

  12. C. Reutenauer, Free Lie Algebras, Clarendon Press, Oxford (1993).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Ulyanovsk State University, Leo Tolstoy 42, Ulyanovsk, 432970, Russia

    V. M. Petrogradsky & A. A. Smirnov

Authors
  1. V. M. Petrogradsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. M. Petrogradsky.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 15, No. 1, pp. 117–124, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petrogradsky, V.M., Smirnov, A.A. On invariants of modular free Lie algebras. J Math Sci 166, 767–772 (2010). https://doi.org/10.1007/s10958-010-9892-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-010-9892-2

Keywords

Search

Navigation