恰无限环与环代数
Just-Infinite Loops and Loop Algebras
- Universidade Federal de Mato Grosso do Sul（南马托格罗索联邦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究拟群及其代数的恰无限性，证明代数恰无限蕴含拟群恰无限，利用Chein构造和广义Moufang双倍建立等价性，并构造满足强性质的例子，同时解释RA拟群的限制。
AI中文摘要：
设 $F$ 为域，$L$ 为拟群（loop）。若 $L$ 为无限且每个非平凡正规子拟群具有有限指数，则称 $L$ 为恰无限（just-infinite）；若可能非结合的拟群代数 $F[L]$ 为无限维且每个非零双边理想具有有限余维数，则称 $F[L]$ 为恰无限。我们首先证明 $F[L]$ 的恰无限性总是蕴含 $L$ 的恰无限性。其次，利用 Chein 构造，我们证明对于每个无限群 $G$，$M(G,2)$ 是恰无限的当且仅当 $G$ 是恰无限的，且 $F[M(G,2)]$ 是恰无限的当且仅当 $F[G]$ 是恰无限的。当 $G$ 为无限非交换群时，我们将此代数等价性推广到广义 Moufang 双倍 $M(G,*,g_0)$。最后，我们构造一个单一的局部有限、剩余有限、非结合 Moufang 拟群 $L$，使得 $F[L]$ 在任意域上都是剩余有限维、局部有限维且恰无限的，并解释为何无限非结合 RA 拟群不能是恰无限的。
英文摘要：
Let $F$ be a field and let $L$ be a loop. We call $L$ just-infinite if it is infinite and every nontrivial normal subloop has finite index, and we call the possibly nonassociative loop algebra $F[L]$ just-infinite if it is infinite-dimensional and every nonzero two-sided ideal has finite codimension. We first prove that just-infiniteness of $F[L]$ always implies just-infiniteness of $L$. Next, using the Chein construction, we show for every infinite group $G$ that $M(G,2)$ is just-infinite if and only if $G$ is just-infinite, and that $F[M(G,2)]$ is just-infinite if and only if $F[G]$ is just-infinite. We extend the algebraic equivalence to the generalized Moufang doubles $M(G,*,g_0)$ whenever $G$ is infinite and nonabelian. Finally, we construct a single locally finite, residually finite, nonassociative Moufang loop $L$ for which $F[L]$ is residually finite-dimensional, locally finite-dimensional, and just-infinite over every field, and we explain why infinite nonassociative RA loops cannot be just-infinite.
