Advertisement

Journal of Mathematical Sciences

, Volume 161, Issue 1, pp 57–69 | Cite as

Nondegenerate graded lie algebras with degenerate transitive subalgebras

  • T. B. Gregory
  • M. I. Kuznetsov
Article

Abstract

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler is analyzed. Transitive irreducible graded Lie algebras \( L = \sum\limits_{i \in \mathbb{Z}} {L_i} \) over an algebraically closed field of characteristic p > 2 with classical reductive component L 0 are considered. We show that if a nondegenerate Lie algebra L containes a transitive degenerate subalgebra L′such that dim L1 > 1, then L is an infinite-dimensional Lie algebra.

Keywords

Minimal Ideal Invariant Bilinear Form Maximal Subalgebra Bracket Operation Recognition Theorem 
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.
    G. M. Benkart and T. B. Gregory, “Graded Lie algebras with classical reductive null component,” Math. Ann., 285, 85–98 (1989).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. M. Benkart, T. B. Gregory, and A. Premet, “The recognition theorem for graded Lie algebras in prime characteristic,” Mem. Amer. Math. Soc. (to appear).Google Scholar
  3. 3.
    G. M. Benkart, A. I. Kostrikin, and M. I. Kuznetsov, “The simple graded Lie algebras of characteristic three with classical reductive component L 0,” Commun. Algebra, 24, 223–234 (1996).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. E. Block, “Modules over differential polynomial rings,” Bull. Amer. Math. Soc., 79, 729–733 (1973).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Brown, “On the structure of some Lie algebras of Kuznetsov,” Michigan Math. J., 39, 85–90 (1992).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. W. Curtis, “Representations of Lie algebras of classical type with applications to linear groups,” J. Math. Mech., 9, 307–326 (1960).MATHMathSciNetGoogle Scholar
  7. 7.
    T. B. Gregory and M. I. Kuznetsov, “On depth-three graded Lie algebras of characteristic three with classical reductive null component,” Commun. Algebra, 33, No. 9, 3339–3371 (2004).CrossRefMathSciNetGoogle Scholar
  8. 8.
    V. G. Kac, “The classification of simple Lie algebras over a field of nonzero characteristic,” Izv. Akad. Nauk SSSR, Ser. Mat., 34, No. 2, 385-408 (1970).MathSciNetGoogle Scholar
  9. 9.
    A. I. Kostrikin and V. V. Ostrik, “On the recognition theorem for Lie algebras of characteristic 3,” Mat. Sb., 186, No. 5, 73–88 (1995).MathSciNetGoogle Scholar
  10. 10.
    A. I. Kostrikin and I. R. Shafarevich, “Graded Lie algebras of finite characteristic,” Izv. Akad. Nauk SSSR, Ser. Mat., 33, No. 2, 251–322 (1969).MathSciNetGoogle Scholar
  11. 11.
    M. I. Kuznetsov, “Truncated induced modules over transitive Lie algebras of characteristic p,” Izv. Akad. Nauk. SSSR, Ser. Mat., 53, No. 3, 557–589 (1989).MATHGoogle Scholar
  12. 12.
    S. M. Skryabin, “New series of simple Lie algebras of characteristic 3,” Mat. Sb., 183, 3–22 (1992).Google Scholar
  13. 13.
    H. Strade, Simple Lie Algebras over Fields of Positive Characteristic. I. Structure Theory, De-Gruyter Expos. Math., Vol. 38, Walter de Gryuter, New York (2004).MATHGoogle Scholar
  14. 14.
    B. Weisfeiler, “On filtered Lie algebras and their associated graded algebras,” Funkts. Anal. Prilozh., 2, No. 1, 94–95 (1968).CrossRefGoogle Scholar
  15. 15.
    B. Weisfeiler, “On the structure of the minimal ideal of some graded Lie algebras in characteristic p > 0,” J. Algebra, 53, No. 2, 344–361 (1978).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • T. B. Gregory
    • 1
  • M. I. Kuznetsov
    • 2
  1. 1.Department of MathematicsThe Ohio State University at MansfieldMansfieldUSA
  2. 2.Nizhny Novgorod State UniversityNizhny NovgorodRussia

Personalised recommendations