On subalgebras of simple Lie algebras of characteristic 𝑝>0

Weisfeiler, B.

doi:10.1090/S0002-9947-1984-0760972-8

On subalgebras of simple Lie algebras of characteristic $p>0$
HTML articles powered by AMS MathViewer

by B. Weisfeiler PDF

Trans. Amer. Math. Soc. 286 (1984), 471-503 Request permission

Abstract:

The main results of the paper are Theorems I.5.1, II.1.3 and III.2.1. Theorem I.5.1 states that if a maximal subalgebra $M$ of a simple finite-dimensional Lie algebra $G$ has solvable quotients of dimension $\geqslant 2$, then every nilpotent element of $H$ acts nilpotently on $G$. Theorem II.1.3 states that if such a simple Lie algebra $G$ contains a maximal subalgebra which is solvable, then $G$ is Zassenbaus-Witt algebra. Theorem III.2.1 states that certain ${\mathbf {Z}}$-graded finite-dimensional simple Lie algebras are either classical or the difference between the number of nonzero positive and negative homogeneous components is large.

References

Richard E. Block, Modules over differential polynomial rings, Bull. Amer. Math. Soc. 79 (1973), 729–733. MR 327844, DOI 10.1090/S0002-9904-1973-13285-9
Richard E. Block, Determination of the differentiably simple rings with a minimal ideal, Ann. of Math. (2) 90 (1969), 433–459. MR 251088, DOI 10.2307/1970745
Marguerite Frank, A new simple Lie algebra of characteristic three, Proc. Amer. Math. Soc. 38 (1973), 43–46. MR 314924, DOI 10.1090/S0002-9939-1973-0314924-0

On the classification of the simple Lie algebras over fields with non-zero characteristic

4

V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1323–1367 (Russian). MR 0259961

Automorphisms of finite order of semi-simple Lie algebras

3

Differential geometry, Lie groups and symmetric spaces

I. L. Kantor, Graded Lie algebras, Trudy Sem. Vektor. Tenzor. Anal. 15 (1970), 227–266 (Russian). MR 0297827

The modular simple Lie algebras with a solvable maximal subalgebra

30

Rimhak Ree, On generalized Witt algebras, Trans. Amer. Math. Soc. 83 (1956), 510–546. MR 82058, DOI 10.1090/S0002-9947-1956-0082058-3
John R. Schue, Cartan decompositions for Lie algebras of prime characteristic, J. Algebra 11 (1969), 25–52. MR 231873, DOI 10.1016/0021-8693(69)90099-4
B. Weisfeiler, On the structure of the minimal ideal of some graded Lie algebras in characteristic $p>0$, J. Algebra 53 (1978), no. 2, 344–361. MR 502633, DOI 10.1016/0021-8693(78)90280-6

The irreducible representations of Lie

algebras

5

Exponentials in Lie algebras of characteristic

5

Robert Lee Wilson, Classification of generalized Witt algebras over algebraically closed fields, Trans. Amer. Math. Soc. 153 (1971), 191–210. MR 316523, DOI 10.1090/S0002-9947-1971-0316523-6
Hans Zassenhaus, The representations of Lie algebras of prime characteristic, Proc. Glasgow Math. Assoc. 2 (1954), 1–36. MR 63359, DOI 10.1017/S2040618500032974