One-and-a-half generation of simple Lie algebras over finite fields: the value k(sl₃(𝔽₂))=3

Abstract

For a simple Lie algebra L over a finite field, Kishnani and Singh define k(L) to be the least k such that every non-zero a ∈ L occurs as the first entry of some generating k-tuple, and observe that no clear answer for k(L) seems to be known; the number is the number of variables in their theorem on images of Lie polynomials. We determine one value: k(sl₃(𝔽₂)) = 3. The upper bound is a generating triple for each of the 255 non-zero elements. The lower bound is structural rather than a report that a search failed: ten proper Lie subalgebras of sl₃(𝔽₂), each containing the root element E₀₁, already cover the algebra, so every pair containing E₀₁ lies inside a proper subalgebra. We also determine the exceptional set exactly. Precisely 49 of the 255 non-zero elements lie in no generating pair, namely the 21 elements with A²=0 and the 28 with A²=A, equivalently the a with ad(a)³=ad(a)²; the remaining 206 each have an explicit partner. The mechanism is characteristic two: if A²=λ A then ad(A)²=λad(A), because the cross term 2AXA vanishes. In particular sl₃(𝔽₂) is 2-generated, as Cantor, Jezernik and Zozaya proved, and is nevertheless not one-and-a-half generated: the two properties come apart at the smallest algebra where they can be compared. All statements are machine-checked in Lean 4.

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