有限维单 Jordan 代数中的幂等词
Idempotent words in finite-dimensional simple Jordan algebras
- University of Zagreb Faculty of Science（萨格勒布大学理学院）
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AI总结：
本文研究特征非2域上有限维单Jordan代数中幂等元的左嵌套积，分类了可表示元素并确定最小词长，发现最大词长随矩阵规模对数增长。
AI中文摘要：
我们研究了特征不同于 2 的域 $\mathbb{F}$ 上有限维单 Jordan 代数中幂等元的有限左嵌套 Jordan 积。在代数闭域上，结构定理将问题归结为标量代数、旋量因子、全矩阵 Jordan 代数 $M_n(\mathbb{F})^+$、对称矩阵代数 $H_n(\mathbb{F})$、辛 Hermitian 代数 $H_{2n}(\mathbb{F},\mathrm{sp})$ 以及分裂 Albert 代数。我们精确确定了哪些元素是可表示的：除 $H_2(\mathbb{F})$ 外，这些是所有非标量元素以及 $0$ 和 $1$，而在 $H_2(\mathbb{F})$ 中，非零平方零元素也不可表示。我们还研究了可表示元素的最小词长。对于任意域上的 $M_n(\mathbb{F})^+$，最大词长随 $n$ 对数增长，并且对于 $n=2$，我们确定了每个非标量矩阵的精确长度。同样的对数增长也适用于 $H_{2n}(\mathbb{F},\mathrm{sp})$。旋量因子和对称矩阵的结果在二次闭域上成立，而全矩阵、辛 Hermitian 和分裂 Albert 的结果在特征不同于 2 的任意域上成立。
英文摘要：
We study finite left-nested Jordan products of idempotents in finite-dimensional simple Jordan algebras over a field $\mathbb{F}$ of characteristic different from $2$. Over an algebraically closed field, the structure theorem reduces the problem to the scalar algebra, spin factors, the full matrix Jordan algebras $M_n(\mathbb{F})^+$, the symmetric matrix algebras $H_n(\mathbb{F})$, the symplectic Hermitian algebras $H_{2n}(\mathbb{F},\mathrm{sp})$, and the split Albert algebra. We determine exactly which elements are representable: except in $H_2(\mathbb{F})$, these are all nonscalar elements together with $0$ and $1$, whereas in $H_2(\mathbb{F})$ the nonzero square-zero elements are also not representable. We also study the minimal word length of representable elements. For $M_n(\mathbb{F})^+$ over an arbitrary field, the maximal word length grows logarithmically in $n$, and for $n=2$ we determine the exact length of every nonscalar matrix. The same logarithmic growth holds for $H_{2n}(\mathbb{F},\mathrm{sp})$. The spin-factor and symmetric-matrix results hold over quadratically closed fields, whereas the full matrix, symplectic Hermitian, and split Albert results hold over arbitrary fields of characteristic different from $2$.
