Complete reducibility of gyrogroup representations

Abstract

© 2019, © 2019 Taylor & Francis Group, LLC. In this article, we show that any finite gyrogroup can be represented on a space of complex-valued functions. In particular, we prove that any linear representation of a finite gyrogroup on a finite-dimensional complex inner product space is unitary and hence is completely reducible using strong connections between linear actions of groups and gyrogroups. Also, we provide an example of a unitary representation of an arbitrary finite gyrogroup, which resembles the group-theoretic left regular representation.