A sum-bracket theorem for simple Lie algebras.

Let g be an algebra over K with a bilinear operation [ ⋅ , ⋅ ] : g × g → g not necessarily associative. For A ⊆ g , let A k 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 = sl n , so n , sp 2 n , e 6 , e 7 , e 8 , f 4 , g 2 : if char (K) is not too small, we have growth of the form | A k | ≥ | A | 1 + ε for all generating symmetric sets A away from subfields of K. Over F p 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 | ≤ | A k | 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. [ABSTRACT FROM AUTHOR]