A sum-bracket theorem for simple Lie algebras

Abstract

Let g be an algebra over K with a bilinear operation [⋅,⋅]:g×g→g not necessarily associative. For A⊆g, let Ak be the set of elements of g written combining k elements of A via + and [⋅,⋅].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+ε 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∩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.