Radicals and embeddings of Moufang loops in alternative loop algebras

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.