About the embedding of Moufang loops in alternative algebras II

TL;DR

This paper investigates the embedding properties of simple Moufang loops derived from matrix Cayley-Dickson algebras, showing that most cannot be embedded into invertible elements of alternative algebras under certain conditions, impacting their structural understanding.

Contribution

It proves that certain simple Moufang loops cannot be embedded into invertible elements of unital alternative algebras when the field characteristic is not 2 and the field is closed under square roots, clarifying their algebraic limitations.

Findings

Simple Moufang loops M(F) are not embeddable into invertible elements of unital alternative algebras under specified conditions.

Embedding is possible for Moufang loops outside the simple class.

Finite Moufang p-loops are shown to be centrally nilpotent.

Abstract

It is known that with precision till isomorphism that only and only loops $M(F) = M_0(F)/<-1>$, where $M_0(F)$ denotes the loop, consisting from elements of all matrix Cayley-Dickson algebra $C(F)$ with norm 1, and $F$ be a subfield of arbitrary fixed algebraically closed field, are simple non-associative Moufang loops. In this paper it is proved that the simple loops $M(F)$ they and only they are not embedded into a loops of invertible elements of any unitaly alternative algebras if $\text{char} F \neq 2$ and $F$ is closed under square root operation. For the remaining Moufang loops such an embedding is possible. Using this embedding it is quite simple to prove the well-known finding: the finite Moufang $p$-loop is centrally nilpotent.