Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.