Abstract
In this article, we examine a subspace L gyr ( 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 L gyr ( 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 ( L gyr ( G ) ) = 1 | γ ( G ) | ∑ ρ ∈ γ ( G ) | Fix ( ρ ) | , where γ ( G ) is the subgroup of Sym ( G ) generated by a class of permutations of G and Fix ( ρ ) = { a ∈ G : ρ ( a ) = a } .
Highlights
G acts on L( G ) by ( a · f )( x ) = f ( a−1 x ), x ∈ G, for all a ∈ G, f ∈ L( G ). This action induces a unitary representation of G on L( G ), known as the left regular representation of G
The reason why we prefer complex-valued functions as a representation space for a finite gyrogroup is indicated in the following theorem
We emphasize that the character of the left regular representation of a finite gyrogroup is not quite as simple as in the case of groups
Summary
G acts on L( G ) by ( a · f )( x ) = f ( a−1 x ), x ∈ G, for all a ∈ G, f ∈ L( G ) This action induces a unitary representation of G on L( G ), known as the left regular representation of G. It seems difficult to extend the notion of group actions to nonassociative structures 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.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
