Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras

A. A. Premet

Sbornik. Mathematics, Tome 66 (1990) no. 2, pp. 555-570

Résumé

Let $\mathfrak G$ be a finite-dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$. It is proved that any two Cartan subalgebras with maximal toral part in $\mathfrak G$ can be obtained from each other by means of a finite chain of elementary transformations that are similar in form to the exponents of the inner root derivations of $\mathfrak G$. The following theorem plays an important role in the proof: Theorem. {\it Let $s$ be a toral rank of $\mathfrak G$ and $e_1,\dots,e_n$ a basis of $\mathfrak G$. There exists $\nu\in\mathbf Z_+$ and homogeneous polynomials $f_0,\dots,f_{s-1},$ in $n$ variables$,$ such that $$ x^{[p^{s+\nu}]}=\sum_{i=0}^{s-1}f_i(x_1,\dots,x_n)x^{[p^{i+\nu}]} $$ $($here $x=x_1e_1+\dots+x_ne_n$ and $\deg f_i=p^{s+\nu}-p^{i+\nu}).$} Bibliography: 16 titles.