Left regular representation of gyrogroups

Abstract

© 2019 by the authors. In this article, we examine a subspace Lgyr(G) of the complex vector space, L(G) = [f: f is a function from G to C], where G is a nonassociative group-like structure called a gyrogroup. The space Lgyr(G) arises as a representation space for G associated with the left regular representation, consisting of complex-valued functions invariant under certain permutations of G. In the case when G is finite, we prove that dim (where γ(G) is the sub group of Sym (G) generated by a class of permutations of G and Fix (ρ) = [a ∈ G: ρ(a) = a].