Lie Algebras Generated by Extremal Elements

Abstract

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals) over a field of characteristic distinct from 2. There is an associative bilinear form on such a Lie algebra; we study its connections with the Killing form. Any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal numbers of extremal generators for the Lie algebras of type A n n ≥ 1 , B n n ≥ 3 , C n n ≥ 2 , D n n ≥ 4 , E n n = 6 7 8 , F 4 and G 2 are shown to be n + 1, n + 1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.

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“…This is a prototypical example of our results. The next smallest case of three generators is treated in [Cohen et al 2001;Zelmanov and Kostrikin 1990] and also by our results below. There the generic Lie algebra is split of type A 2 and more interesting degenerations exist.…”

“…Let V be a finite-dimensional homogeneous subspace of Ᏺ such that Ᏺ = V ⊕Ᏽ(0); such a subspace exists as ᏸ(0) is finite-dimensional [Cohen et al 2001;Zelmanov and Kostrikin 1990] and Ᏽ(0) is homogeneous. Note that the theorem only requires that Ᏺ = V + Ᏽ(0); we will argue later why this suffices.…”

“…The extremal form. Cohen et al [2001] proved that on any Lie algebra over a field of characteristic unequal to 2 that is generated by finitely many extremal elements, there is a unique bilinear form κ such that x x y = κ(x, y)x for all extremal x. Moreover, it is shown there that κ is symmetric and associative: κ(x, y) = κ(y, x) and κ(x y, z) = κ(x, yz) for all x, y, z.…”

“…The prominence of extremal elements in the work of Kostrikin and Zel'manov and in modular Lie algebra theory led to the natural problem of describing all Lie algebras generated by a fixed number of extremal elements [Cohen et al 2001;in 't panhuis et al 2007;Postma 2007;Roozemond 2005].…”

“…For general f ∈ (Ᏺ * ) it turns out that dim ᏸ( f ) ≤ dim ᏸ(0); see [Cohen et al 2001] or the proof of Theorem 1 below. It is therefore natural to focus on the Lie algebras ᏸ( f ) of the maximal possible dimension dim ᏸ(0).…”