Just-Infinite Loops and Loop Algebras

Abstract

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 $M(G,\ast ,g_0)$ 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.