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The classification problem for simple lie algebras of characteristic p

Part of the Lecture Notes in Mathematics book series (LNM,volume 933)

Keywords

  • Classical Type
  • Divided Power
  • Minimal Ideal
  • Root Space
  • Trace Form

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.

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References

  • [Bn] Benkart, G. M.: On inner ideals and ad-nilpotent elements of Lie algebras. Trans. Amer. Math. Soc. 232 (1977), 61–81.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [B-I] Benkart, G. M., Isaacs, I. M.: Lie algebras with nilpotent centralizers. Canad. J. Math. 31 (1979), 929–941.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [B-I-O] Benkart, G. M., Isaacs, I. M., Osborn, J. M.: Lie algebras with self-centralizing ad-nilpotent elements. J. Algebra 57 (1979), 279–309.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Bl 1] Block, R. E.: Trace forms on Lie algebras. Can. J. Math. 14 (1962), 553–564.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Bl 2] Block, R. E.: On the Mills-Seligman axioms for Lie algebras of classical type. Trans. Amer. Math. Soc. 121 (1966), 378–392.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Bl 3] Block, R. E.: Determination of the differentiably simple rings with a minimal ideal. Ann. of Math. (2) 90 (1969), 433–459.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Bl 4] Block, R. E.: Irreducible representations of Lie algebra extensions. Bull. Amer. Math. Soc. 8 (1974), 868–872.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Bl 5] Block, R. E.: Restricted simple Lie algebras of rank 2 or of toral rank 2. 1981 Yale Conference Proceedings, Contemporary Math., Amer. Math. Soc. (In press).

    Google Scholar 

  • [B-W 1] Block, R. E., Wilson, R. L.: On filtered Lie algebras and divided power algebras. Comm. in Algebra 3 (1975), 571–589.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [B-W 2] Block, R. E., Wilson, R. L.: The simple Lie p-algebras of rank two. Ann of Math. (In press).

    Google Scholar 

  • [Dm 1] Demuskin, S. P.: Cartan subalgebras of the simple Lie p-algebras Wn and Sn (Russian). Sibirsk. Mat. Z. 11 (1970) 310–325; English transl., Siberian Math. J. 11 (1970) 233–245.

    MathSciNet  MATH  Google Scholar 

  • [Dm 2] Demuskin, S. P.: Cartan subalgebras of simple nonclassical Lie p-algebras (Russian). Izv. Akad. Nauk SSSR Ser. Math. 36 (1972), 915–932; English transl., Math. USSR-Izv. 6 (1972), 905–924.

    MathSciNet  Google Scholar 

  • [El] El’sting, G. O.: Izv. Vysš. Učebn. Zaved. Matematika 4 (1981), 64–68.

    MathSciNet  Google Scholar 

  • [Er] Ermolaev, Yu. B.: Izv. Vysš. Učebn. Zaved. Matematika 7 (1981), 80–84, 8 (1981) 66–70, 8 (1981), 70–74.

    MathSciNet  Google Scholar 

  • [Gr 1] Gregory, T. B.: Simple Lie algebras with classical reductive null component. J. Algebra 63 (1980), 484–493.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Gr 2] Gregory, T. B.: A characterization of the contact Lie algebras. Proc. Amer. Math. Soc. 82 (1981), 505–511.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Gr 3] Gregory, T. B.: On simple reducible Lie algebras of depth two. Proc. Amer. Math. Soc. 83 (1981), 31–35.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Jc] Jacobs, J. B.: On classifying simple Lie algebras of prime characteristic by nilpotent elements. J. Algebra 19 (1971), 31–50.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Kc 1] Kac, V. G.: The classification of the simple Lie algebras over a field with non-zero characteristic (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 385–408; English transl., Math. USSR-Izv. 4 (1970), 391–413.

    MathSciNet  Google Scholar 

  • [Kc 2] Kac, V. G.: Description of filtered Lie algebras with which grade Lie algebras of Cartan type are associated (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 800–838; Errata 40 (1976), 1415; English transl. Math. USSR-Iva, 8 (1974), 801–835; Errata, 10 (1976), 1339.

    MathSciNet  Google Scholar 

  • [Kp 1] Kaplansky, I.: Lie algebras of characreristic p. Trans. Amer. Math. Soc. 89 (1958), 149–183.

    MathSciNet  MATH  Google Scholar 

  • [Kp 2] Kaplansky, I.: Lie algebras and locally compact groups. University of Chicago Press (1971).

    Google Scholar 

  • [Ks 1] Kostrikin, A. I.: Simple Lie p-algebras (Russian). Trudy Mat. Inst. Steklov 64 (1961), 78–89; English transl., Amer. Math. Soc. Transl. Ser. (2) 55 (1966), 195–206.

    MathSciNet  MATH  Google Scholar 

  • [Ks 2] Kostrikin, A. I.: Squares of adjoint endomorphisms in simple Lie p-algebras (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 445–487; English transl., Math. USSR-Izv. 1 (1967), 434–473.

    MathSciNet  MATH  Google Scholar 

  • [Ks 3] Kostrikin, A. I.: Variations modulaires sur un theme de Cartan. Actes. Congres Intern. Math. Tome 1 (1970) 285–292.

    MathSciNet  Google Scholar 

  • [K-S 1] Kostrikin, A. I., Šafarevič, I. R.: Cartan pseudogroups and Lie p-algebras (Russian). Dokl. Akad. Naud SSSR 168 (1966), 740–742; English transl., Soviet Math. Dokl. 7 (1966), 715–718.

    Google Scholar 

  • [K-S 2] Kostrikin, A. I., Šafarevič, I. R.: Graded Lie algebras of finite characteristic (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 251–322; English transl., Math. USSR-Irv. 3 (1969), 237–304.

    MathSciNet  Google Scholar 

  • [Kr] Krylyuk, Ja. C.: Uspehi Mat. Nauk (1979), 203–204.

    Google Scholar 

  • [Kz 1] Kuznetsov, M. K.: Simple modular Lie algebras with a solvable maximal subalgebra (Russian). Mat. Sbornik 101 (1976), 77–86.

    MATH  Google Scholar 

  • [Kz 2] Kuznetsov, M. K.: Mat. Sb. (To appear).

    Google Scholar 

  • [Kz 3] Kuznetsov, M. K.: (To appear).

    Google Scholar 

  • [M-S] Mills, W. M., Seligman, G. B.: Lie algebras of classical type. J. Math. Mech. 6 (1957), 519–548.

    MathSciNet  MATH  Google Scholar 

  • [Sc] Schue, J. R.: Cartan decompositions for Lie algebras of prime characteristic. J. Algebra 11 (1969), 25–52; Erratum 13 (1969), 558.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Sl 1] Seligman, G. B.: On Lie algebras of prime characteristic. Mem. Amer. Math. Soc. 19 (1956).

    Google Scholar 

  • [Sl 2] Seligman, G. B.: Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40. Springer-Verlag New York, Inc., New York, (1967).

    Google Scholar 

  • [Sr] Serconek, S.: Finite Cartan matrices and Lie algebras of classical type. J. Algebra.

    Google Scholar 

  • [St] Strade, H.: Nonclassical simple Lie algebras and strong degeneration. Arch. Math. (Basel) 24 (1973), 482–485.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Ty] Tyurin, S. A.: Mat. Zametki (1978).

    Google Scholar 

  • [Ws 1] Weisfeiler, B. Ju.: On the structure of the minimal ideal of some graded Lie algebras in characteristic p>0. J. Algebra 53 (1978), 344–361.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Ws 2] Weisfeiler, B. Ju.: (To appear).

    Google Scholar 

  • [W-K] Weisfeiler, B. Ju., Kac, V. G.: Exponentials in Lie algebras of characteristic p (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 762–788; English transl., Math. USSR-Izv. 5 (1971), 777–803.

    MathSciNet  MATH  Google Scholar 

  • [Wl 1] Wilson, R. L.: Classification of generalized Witt algebras over algebraically closed fields. Trans. Amer. Math. Soc. 153 (1971), 191–210.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wl 2] Wilson, R. L.: The roots of a simple Lie algebra are linear. Bull. Amer. Math. Soc. 82 (1976), 607–608.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wl 3] Wilson, R. L.: A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic. J. Algebra 40 (1976), 418–465.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wl 4] Wilson, R. L.: Simple Lie algebras of toral rank one. Trans. Amer. Math. Soc. 236 (1978), 287–295.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wl 5] Wilson, R. L.: Simple Lie algebras of type S. J. Algebra 62 (1980), 292–298.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wl 6] Wilson, R. L.: A “new” proof of Kaplansky’s theorem on simple Lie algebras of rank one. Algebra Carbondale 1980, Proceedings, Lect. Notes in Math. 848, Springer-Verlag (1981), 1–12.

    CrossRef  Google Scholar 

  • [Wl 7] Wilson, R. L.: The classification problem for simple Lie algebras of characteristic p. Algebra Carbondale 1980, Proceedings, Lect. Notes in Math. 848, Springer-Verlag (1981), 13–32.

    CrossRef  MathSciNet  Google Scholar 

  • [Wl 8] Wilson, R. L.: Restricted simple Lie algebras with toral Cartan subalgebras. 1981 Yale Conference Proceedings, Contemporary Math., Amer. Math. Soc. (In press).

    Google Scholar 

  • [Wn 1] Winter, D. J.: On the toral structure of Lie p-algebras. Acta Math. 123 (1969), 70–81.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wn 2] Winter, D. J.: Triangulable subalgebras of Lie p-algebras. Pac. J. Math. 81 (1979), 273–281.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wn 3] Winter, D. J.: Cartan decompositions and Engel subalgebra triangulability. J. Algebra 62 (1980), 400–417.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Wn 4] Winter, D. J.: A combinatorial theory of symmetry and applications to Lie algebras. Algebra Carbondale 1980, Proceedings, Lect. Notes in Math. 848, Springer-Verlag (1981), 41–62.

    CrossRef  MathSciNet  Google Scholar 

  • [Wn 5] Winter, D. J.: Symmetrysets, J. Alg., 73 (1981), 238–247.

    CrossRef  MathSciNet  MATH  Google Scholar 

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Block, R.E. (1982). The classification problem for simple lie algebras of characteristic p. In: Winter, D. (eds) Lie Algebras and Related Topics. Lecture Notes in Mathematics, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093351

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