On Cartan subalgebras of Lie $p$-algebras

A. A. Premet

Izvestiya. Mathematics , Tome 29 (1987) no. 1, pp. 145-157

Résumé

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.