Geodesic
On Cartan subalgebras of Lie $p$-algebras
Izvestiya. Mathematics , Tome 29 (1987) no. 1, pp. 145-157.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown, using the technique of switching toral subalgebras, that in finitedimensional Lie $p$-algebras every Cartan subalgebra with maximal toral part has dimension equal to the rank of the algebra. As is known, every Cartan subalgebra of a Lie $p$-algebra $\mathfrak g$ is of the form $\mathfrak g_x^0$, where $\mathfrak g_x^0$ is the nilspace of the endomorphism $\operatorname{ad}x$, $x\in\mathfrak g$. It is proved that there exists a Zariski-open subset $V\subset\mathfrak g$ such that for every $x\in V$ the subspace $\mathfrak g_x^0$ is a Cartan subalgebra with maximal toral part. A further result is the proof that the class of Cartan subalgebras with maximal toral part is the same as the class of Cartan subalgebras with minimal nilpotent part. The results are used to settle a question concerning anisotropic forms of Lie algebras over finite fields. Bibliography: 12 titles. 
@article{IM2_1987_29_1_a8,
     author = {A. A. Premet},
     title = {On {Cartan} subalgebras of {Lie} $p$-algebras},
     journal = {Izvestiya. Mathematics },
     pages = {145--157},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_29_1_a8/}
}
TY  - JOUR
AU  - A. A. Premet
TI  - On Cartan subalgebras of Lie $p$-algebras
JO  - Izvestiya. Mathematics 
PY  - 1987
SP  - 145
EP  - 157
VL  - 29
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1987_29_1_a8/
LA  - en
ID  - IM2_1987_29_1_a8
ER  - 
%0 Journal Article
%A A. A. Premet
%T On Cartan subalgebras of Lie $p$-algebras
%J Izvestiya. Mathematics 
%D 1987
%P 145-157
%V 29
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1987_29_1_a8/
%G en
%F IM2_1987_29_1_a8
A. A. Premet. On Cartan subalgebras of Lie $p$-algebras. Izvestiya. Mathematics , Tome 29 (1987) no. 1, pp. 145-157. http://geodesic.mathdoc.fr/item/IM2_1987_29_1_a8/

[1] Burbaki N., Algebra. Mnogochleny i polya. Uporyadochennye gruppy, Nauka, M., 1965 | MR

[2] Burbaki N., Gruppy i algebry Li, gl. 1–3, Mir, M., 1976 | MR

[3] Burbaki N., Gruppy i algebry Li, gl. 7,8, Mir, M., 1978 | MR

[4] Dzhekobson N., Algebry Li, Mir, M., 1964 | MR

[5] Borel A., Lineinye algebraicheskie gruppy, Mir, M., 1972 | MR | Zbl

[6] Serr Zh.-P., Kogomologii Galua, Mir, M., 1968 | MR

[7] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR | Zbl

[8] Winter D. J., “On the toral structure of Lie $p$-algebras”, Acta Math., 123 (1969), 70–81 | DOI | MR

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

[10] Block R. E., Wilson R. L., “The simple Lie $p$-algebras of rank two”, Ann. Math., 115:1 (1982), 93–168 | DOI | MR | Zbl

[11] Wilson R. L., “Classification of the restricted simple Lie algebras with toral Cartan subalgebras”, Algebra, 83:2 (1983), 531–570 | DOI | MR | Zbl

[12] Farnsteiner R., “On ad-semisimple Lie algebras”, Algebra, 83:2 (1983), 510–519 | DOI | MR | Zbl