Left Regular Representation of Gyrogroups

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 } .

Gyrogroup through its Grothendieck Group Completion and Right gyrogroup action

“…In the continuation of this, Suksumran [5] explored the notion of gyrogroup actions which turns out to be a natural generalization of the usual notion of group action. Later, the author [6,7,8] In this article, we show that this is equivalent to having an action of the group M(G) on X and a linear representation of M(G) on V , where M(G) is the Grothendieck group completion of G. Indeed, the category of gyrogroup actions is equivalent to the category of group actions and the category of gyrogroup representations is equivalent to the category of group representations. As such, most of the results are deducible from the known results for groups by exploring the quotient map ν : G −→ M(G).…”

“…Let (6) follows from ( 2) and ( 5) by taking (7) follows from (3). Now we will give the proof of identity (8).…”

“…] on G is the same as representation (action) defined in [5] ( [8]). Also, from every representation (action) of [R(G)], we have a representation (action) of G. (7) Let H be a L-subgyrogroup of G. Then ν(H) be the subgroup of M(G).…”