Introduction to Generalized Digroups - Second Isomorphism Theorem
Maldonado E and Vergara G
Published on: 2021-04-26
In this article we will study in detail the theory on generalized digroups recently developed by O. Salazar, R. Velásquez and L. Wills and that they published in Salazar, Velasquez & wills (2016); for this, we initially present the basic and introductory theory on the subject, its main characterizations, theorems and some examples. Then we will present a version of the first ismophisms theorem for generalized digroups, which in turn will allow to de- velop the second isomorphism theorem under this context. Finally, concrete examples are exhibited.
KeywordsGeneralized digroups; Isomorphisms; Abstract algebra
Tangential algebra to a Lie group is a structure known as Lie algebra. Loday , introduced the notion of Leibniz algebra as a generalization of Lie algebras, wondered if it would be possible to guarantee the existence of an algebraic structure that generalized the concept of Lie group and whose tangent space it was his corresponding algebra of Leibniz. Although it achieved some significant progress, it could not determine exactly what properties this generalization should fulfill and a little frustrated by the elusiveness of said structure, it ended up calling it "Coquecigrue" (a mythological creature).
Later, independently and motivated by Loday’s previous work and the problem of "Coquecigrue", Kinyon, Felipe and Liu introduced the notion of digroups, also finding a partial solution for a particular type of Leibniz algebras, called separable Leibniz algebras. In the case of non-separable ones, the problem remains open [2-4].
In particular, Kinyon showed that each digroup is the product of a group with a trivial digroup. For its part, the partial solution to the Coquecigrue problem is based on the fact that for each Leibniz algebra that is separable over an ideal that contains the ideal generated by the squares, there is a special type of Lie digroup whose tangential algebra is isomorphic to the algebra of Leibniz given.
Taking as reference the previous considerations, in this article we will study the generalized digroups, which are a larger 1class of digroups than the one introduced by Kinyon, Felipe and Liu. It is also noteworthy that, this new structure avoids the presumption of the existence of unique bilateral inverses with respect to a fixed bar unit and only assumes the existence of lateral inverses that should not necessarily be equal, which makes them less restrictive.
The work consists in the first instance, in exhibiting the theory developed by Salazar, Velasquez & wills . For this we present the generalized digroups as an algebraic structure that generalizes the classic concept of digroup, its basic properties, characterizations and the notion of the First Theorem of Isomorphisms. Then, following procedures analogous to those used in group theory, we will intro- duce the subdigroups, the concept of normality, product, quotient and the version of the Second Isomorphism Theorem for generalized digroups. Finally, illustrative examples and some conclusions are presented.