Let $\mathfrak{ g}$ be an algebra over $K$ with a bilinear operation $[\cdot,\cdot]:\mathfrak{ g}\times\mathfrak{ g}\rightarrow\mathfrak{ g}$ not necessarily associative. For $A\subseteq\mathfrak{ g}$, let $A^{ k}$ be the set of elements of $\mathfrak{ g}$ written combining $k$ elements of $A$ via $+$ and $[\cdot,\cdot]$. We show a "sum-bracket theorem" for simple Lie algebras over $K$ of the form $\mathfrak{ g}=\mathfrak{ sl}_{ n},\mathfrak{ so}_{ n},\mathfrak{ sp}_{ 2n},\mathfrak{ e}_{ 6},\mathfrak{ e}_{ 7},\mathfrak{ e}_{ 8},\mathfrak{ f}_{ 4},\mathfrak{ g}_{ 2}$: if $\mathrm{ char}(K)$ is not too small, we have growth of the form $|A^{ k}|\geq|A|^{ 1+\varepsilon}$ for all generating symmetric sets $A$ away from subfields of $K$. Over $\mathbb{ F}_{ p}$ in particular, we have a diameter bound matching the best analogous bounds for groups of Lie type [BDH21]. As an independent intermediate result, we prove also an estimate of the form $|A\cap V|\leq|A^{ k}|^{ \dim(V)/\dim(\mathfrak{ g})}$ for linear affine subspaces $V$ of $\mathfrak{ g}$. 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.
Categories: math.RA