A sum-bracket theorem for simple Lie algebras

Let g be an algebra over K with a bilinear operation [& BULL;, & BULL;] : g x g -g not necessarily associative. For A C g, let Ak be the set of elements of g written combining k elements of A via + and [& BULL;, & BULL;].We show a "sum-bracket theorem" for simple Lie algebras over K of the form g = sln,son, sp2n, e6, e7, e8, f4, g2: if char(K) is not too small, we have growth of the form |Ak| ? |A|1+& epsilon; for all generating symmetric sets A away from subfields of K. Over Fp in particular, we have a diameter bound matching the best analogous bounds for groups of Lie type [2]. As an independent intermediate result, we prove also an estimate of the form |A n V | < |Ak|dim(V )/ dim(g) for linear affine subspaces V of g. This estimate is valid for all simple algebras, and k is especially small for a large class of them including associative, Lie, and Mal'cev algebras, and Lie superalgebras.& COPY; 2023 Elsevier Inc. All rights reserved.