Cartan Subalgebras of Zassenhaus Algebras

Abstract

Cartan subalgebras play a very important role in the classification of the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic zero. It is well-known [5, 273] that any two Cartan subalgebras of such an algebra are conjugate, i.e. images of one another under some automorphism of the algebra. On the other hand, there exist finitedimensional simple Lie algebras over fields of finite characteristic p possessing non-conjugate Cartan subalgebras [2; 3; 4]. The simple Lie algebras discovered by Zassenhaus [6] also possess non-conjugate Cartan subalgebras, and we shall give a complete classification of Cartan subalgebras of these algebras in this paper.

Simple Lie Algebras of Toral Rank One

“…Note that the toral rank does depend on the choice of H. For example, Brown [2] has shown that the toral rank of W(l : n) with respect to H is either 1 or n depending on the choice of H.…”

The nilpotent variety of W(1;n)p is irreducible

“…This is a finite field of q elements. Then L has a k-basis {e α | α ∈ F q } with the Lie bracket given by [e α , e β ] = (β − α)e α+β [11, Theorem 7.6.3 (1)].…”

“…It is well known that L is a free O(1; n)-module of rank 1 generated by the special derivation ∂ such that ∂(x (a) ) = x (a−1) if 1 ≤ a ≤ p n − 1 and 0 otherwise [10, Ch. 4, Proposition 2.2 (1)]. When n = 1, L coincides with the Witt algebra W (1; 1) := Der(O(1; 1)), a simple and restricted Lie algebra.…”

“…When n ≥ 2, L provides the first example of a simple, non-restricted Lie algebra [10,Ch. 4, Theorem 2.4 (1)]. From now on we always assume that n ≥ 2.…”

Simple Lie algebras of characteristic 𝑝 with dependent roots

“…Brown [Br] has shown that a Cartan subalgebra of W(l : n) is one of the following: (i) a Cartan subalgebra of dimension pn~ contained in a subalgebra of codimension one, or (ii) a one-dimensional Cartan subalgebra. In the second case there is a basis {ufo e fi} for some additive subgroup fi of F with H = Fu0 , and with multiplication given by \ua, uf\ = (x -o)ua+T (see [Br,Theorem 4]). By Corollary 3.4 of [BIO], the subalgebra of W(l : n) of codimension one is unique, and hence it is the solvable maximal subalgebra M0 spanned by the elements v for / > 0 as in (2.5).…”