Radicals and embeddings of Moufang loops in alternative loop algebras

TL;DR

This paper introduces the concept of alternative loop algebras for Moufang loops, defines radicals, and proves an analogue of Wedderburn's theorem, showing embedding properties of loops within these algebraic structures.

Contribution

It establishes the radicals for classes of alternative loop algebras and Moufang loops, and proves an analogue of Wedderburn's theorem for finite-dimensional cases.

Findings

Radicals R and S are defined for the classes of algebras and loops.

An analogue of Wedderburn's theorem is proved for finite-dimensional cases.

Loops in the radical class can be embedded into invertible elements of their algebra.

Abstract

The paper defines the notion of alternative loop algebra F[Q] for any nonassociative Moufang loop Q as being any non-zero homomorphic image of the loop algebra FQ of a loop Q over a field F. For the class M of all nonassociative alternative loop algebras F[Q] and for the class L of all nonassociative Moufang loops Q are defined the radicals R and S, respectively. Moreover, for classes M, L is proved an analogue of Wedderburn Theorem for finite dimensional associative algebras. It is also proved that any Moufang loop Q from the radical class R can be embedded into the loop of invertible elements U(F[Q])of alternative loop algebra F[Q]. The remaining loops in the class of all nonassociative Moufang loops L cannot be embedded into loops of invertible elements of any unital alternative algebras.