The structure of g-digroup actions and representation theory

Abstract

The aim of this paper is to propose two possible ways of defining a g-digroup action and a first approximation to representation theory of g-digroups.

A geometrical approach to g-digroups

“…In the paper [21] H. Guzmán and F. Ongay proposed and studied a definition of digroup action on sets and in [1] such concept was extended to g-digroups. For this reason, it will be noted that many of the results exposed in this section are motivated by the paper [21].…”

“…In a similar way we have Burnside's formula for g-digroup actions. Since its proof is similar to the one given in group theory and in [1], we will omit it. Proof Let (g, e) ∈ φ l (Stab D,◁ (m)) ⊂ G ξ l × E. Then, there is x ∈ Stab D,◁ (m), with (g, e) = φ l (x).…”

“…It is well known that representation theory plays a relevant role in order to solve and interpret problems in group theory, this also provides a manner to see the elements of a group as geometric symmetries, which allows to describe aspects that are unchanged under some transformations. Following this line of reasoning, in this paper we present a deep study of the concept of generalized g-digroup action previously introduced in [1]. Thus, we prove a version of the Orbit stabilizer theorem, see Corollary 2, the class equation, see Theorem 7, Burnside's formula, see Theorem 8, and we connect g-digroup actions with trioids theory, see [15], like the connection of group action with g-digroups given in [9].…”

“…

Representation group theory plays an important role in group theory, much more in geometry, on which we can determine, in many cases, what kind of geometry or geometries a manifold M supports by knowing a faithful representation of the fundamental group Π1(M ) in the group Iso + (M ) of preserving-orientations isometries of M . Motivated by this and by the definition of a g-digroup action given in [1], in this paper we present a study of such action by proving some interesting results that are well known in group actions, for example: the orbit stabilizer theorem, the class equation, Burnside's formula and we relate g-digroup actions with trioids. We introduce the definition of representation of a g-digroup into dialgebras and a strong connection with g-digroup action, like in the case of group.

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“…A partial answer is given in the following proposition. Necessary and sufficient conditions are given in Theorems 10 and 11 of [1].…”