A sum-bracket theorem for simple Lie algebras

Abstract

Let $\mathfrak{g}$ be an algebra over $K$ with a bilinear operation $[\cdot ,\cdot ]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ 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 $|A^k|\ge |A{|}^{1+\epsilon }$ 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 $|A\cap V|\le |A^k{|}^{\dim (V)/\dim (\mathfrak{g})}$ 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.

36 pages; v2: restructured introduction, minor edits, final version

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• New results on sum-product type growth over fields
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