Averaging operators on groups, racks and Leibniz algebras

This paper considers averaging operators on various algebraic structures and studies the induced structures. We first introduce the notion of an averaging operator on a group $G$ and show that it induces a rack structure. Moreover, the given group structure and the induced rack structure form a group-rack. We observe that any pointed group-rack can be embedded into an averaging group. We show that the differentiation of a smooth pointed averaging operator on a Lie group gives rise to an averaging operator on the corresponding Lie algebra. Next, we consider averaging operators on a rack that induces a hierarchy of new rack structures. Moreover, any two racks with increasing hierarchy levels form a rack-pairing, a structure that is related to two-sided skew braces by conjugation. We also consider averaging operators on cocommutative Hopf algebras and braided vector spaces in relations to averaging operators on groups, Lie algebras and racks. In the end, we define averaging operators on a Leibniz algebra, find the induced structure and show that the differentiation of a smooth pointed averaging operator on a pointed Lie rack yields an averaging operator on the corresponding Leibniz algebra.