Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Left Regular Representation of Gyrogroups

“…Gyrogroups are nonassociative algebraic structures that share many properties with groups [1,2,4,6,7]. One of the main aspects of gyrogroup structures is that the Cayley table of a finite gyrogroup represents a 'Latin square'-an object studied in combinatorics and in experimental design.…”

“…For more details, we refer the reader to [1]. In [2], the authors extend the study of gyrogroup actions and representations to a linear version. A gyrogroup action of G on a vector space V is linear if in addition every element of G induces an (invertible) linear transformation on V; that is, if the map…”

“…Furthermore, ϕ is irreducible if the invariant subspaces of V are only the trivial subspace and the whole space. For more details, we refer the reader to [2].…”

“…Hence, it seems difficult to extend the notion of group actions to nonassociative structures. However, this can be done (in certain circumstances) by employing the equivalence of group actions and permutation representations, see, for instance, [1][2][3][4]. This motivates us to study the left regular representation in the setting of gyrogroups in more detail.…”

On Metric Structures of Normed Gyrogroups

“…In [27], Ungar studies a parametrization of the Lorentz transformation group. This leads to the formation of gyrogroup theory, a rich subject in mathematics [3,9,[23][24][25][26]30]. Loosely speaking, a gyrogroup is a group-like structure in which arXiv:1810.10491v2 [math.MG] 3 Nov 2018 the associative law fails to satisfy.…”