论文标题
简单谎言代数的汇总定理
A sum-bracket theorem for simple Lie algebras
论文作者
论文摘要
让$ \ mathfrak {g} $成为$ k $的代数，带有双线操作$ [\ cdot，\ cdot]：\ mathfrak {g} \ times \ times \ times \ mathfrak {g} \ rightArrow \ rightArrow \ mathfrak \ mathfrak {g} $不一定关联。对于$ a \ subseteq \ mathfrak {g} $，让$ a^{k} $是$ \ mathfrak {g} $写入$ a $ a $ a $ a $ a $ a $ a $和$ [\ cdot，\ cdot，\ cdot] $的元素的集合。 我们显示一个“ sum-bracket定理”，用于表格的$ k $以上的简单谎言代数$ \ mathfrak {g} = \ mathfrak {sl} _ {n}，\ mathfrak {so} _ {n}，\ mathfrak {sp} _ {2n}，\ mathfra K {如果$ \ mathrm {char}（k）$不太小，则我们的形式$ | a^{k} | \ geq | \ geq | a |^{1+ \ varepsilon} $，用于所有生成对称的对称集$ a $ a $ a a $ ak of $ k $的子场。超过$ \ mathbb {f} _ {p} $，我们的直径界限匹配了谎言类型组的最佳类似界限[bdh21]。 作为一个独立的中间结果，我们还证明了表格$ | a \ cap v | \ leq | a^{k} |^{\ dim（v）/\ dim（\ mathfrak {g}）} $的线性仿射子序列$ v $ $ \ mathfrak {g} $。该估算对于所有简单的代数都有效，对于包括联想，谎言和Mal'Cev代数以及谎言Superalgebras在内的大量级别，$ k $特别小。
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.
