Geodesic
New methods for the classification of the simple modular Lie algebras
Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 235-245 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We investigate the structure of simple modular Lie algebras $L$ over an algebraically closed field of characteristic $p>7$. Let $T$ denote an optimal torus in some $p$-envelope $L_p$. We prove: If $Q(L,T)=L$ and $C_L(T)$ is a Cartan subalgebra, then $L$ is classical. If $Q(L,T)\ne L$ and $C_L(T)$ distinguishes the roots of $T$ on $L/Q(L,T)\ne 0$, then $L$ is of Cartan type. The methods give new proofs even for the restricted simple Lie algebras. 
@article{SM_1992_71_1_a13,
     author = {H. Strade},
     title = {New methods for the classification of the simple modular {Lie} algebras},
     journal = {Sbornik. Mathematics},
     pages = {235--245},
     year = {1992},
     volume = {71},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_71_1_a13/}
}
TY  - JOUR
AU  - H. Strade
TI  - New methods for the classification of the simple modular Lie algebras
JO  - Sbornik. Mathematics
PY  - 1992
SP  - 235
EP  - 245
VL  - 71
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1992_71_1_a13/
LA  - en
ID  - SM_1992_71_1_a13
ER  - 
%0 Journal Article
%A H. Strade
%T New methods for the classification of the simple modular Lie algebras
%J Sbornik. Mathematics
%D 1992
%P 235-245
%V 71
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1992_71_1_a13/
%G en
%F SM_1992_71_1_a13
H. Strade. New methods for the classification of the simple modular Lie algebras. Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 235-245. http://geodesic.mathdoc.fr/item/SM_1992_71_1_a13/

[1] Kostrikin A. I., “Odna teorema o poluprostykh $p$-algebrakh Li”, Matem. zametki, 2:5 (1967), 465–474 | MR | Zbl

[2] Kostrikin A. I., Shafarevich I. R., “Psevdogruppy Kartana i $p$-algebry Li”, DAN SSSR, 168:4 (1966), 740–742 | MR

[3] Kostrikin A. I., Shafarevich I. R., “Graduirovannye algebry Li konechnoi kharakteristiki”, Izv. AN SSSR. Ser. matem., 33:2 (1969), 251–322 | MR | Zbl

[4] Benkart G. M., Simple modular Lie algebras with 1-sections that are classical or solvable, Preprint | MR | Zbl

[5] Benkart G. M., Simple modular Lie algebras with a 1-section that is Witt or Hamiltonian, Preprint | Zbl

[6] Benkart G. M., Gregory T., “Graded Lie algebras with classical reductive null component”, Math. Ann. | MR | Zbl

[7] Benkart G. M., Osborn J. M., Strade H., Construction of subalgebras in simple modular Lie algebras, Preprint | Zbl

[8] Block R. E., Wilson R. L., “The simple Lie $p$-algebras of rank two”, Annals Math., 115 (1982), 93–168 | DOI | MR | Zbl

[9] Block R. E., Wilson R. L., “Classification of the restricted simple Lie algebras”, J. Algebra, 114 (1988), 115–259 | DOI | MR | Zbl

[10] Seligman G. B., Modular Lie algebras, Ergeb. d. Math. u. ihrer Grenzb, 40, Springer-Verlag, Berlin, 1967 | MR | Zbl

[11] Strade H., “The absolute toral rank of a Lie algebra”, Lie algebras (Madiso 1987), Lect. Notes Math., 1373, eds. G. M. Benkart. J. M. Osborn, Springer, N. Y.–Berlin, 1989, 1–28 | MR

[12] Strade H., “The classification of the simple modular Lie algebras. Determination of the two-sections”, Ann. Math., 130 (1989), 643–677 | DOI | MR | Zbl

[13] Strade H., “The classification of the simple modular Lie algebras. II: The toral structure” (to appear)

[14] Strade H., “The classification of the simple modular Lie algebras. III: Solution of the classical case” (to appear)

[15] Strade H., The classification of the simple modular Lie algebras. IV: Solution of the nonhamiltonian case, Preprint | MR

[16] Strade H., Farnsteiner R., Modular Lie algebras and their representations, Marcell Dekker Textbooks and Monographs, 116, 1988 | MR | Zbl

[17] Wilson R. L., “Cartan subalgebras of simple Lie algebras”, Trans. Amer. Math. Soc., 234 (1977), 435–446 | DOI | MR | Zbl