A sum-bracket theorem for simple Lie algebras

TL;DR

This paper establishes a sum-bracket growth theorem for simple Lie algebras over various fields, showing that generating sets grow significantly under algebraic operations, with implications for diameter bounds and subset intersection estimates.

Contribution

It introduces a sum-bracket theorem for simple Lie algebras, providing new growth bounds and intersection estimates applicable across different algebraic structures.

Findings

Growth of generating sets in simple Lie algebras is at least polynomially larger after a few operations.

Diameter bounds for Lie-type groups over finite fields match the best known results.

Intersection estimates for subsets with linear subspaces are established for all simple algebras.

Abstract

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.