Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras

Algorithm to compute abelian subalgebras and ideals in Malcev algebras

“…Let us note that the algorithm presented here consists of the theoretical adaptation and technical translation into the case of Malcev algebras from that introduced in [3] for Lie algebras and [4] for Leibniz ones. In this sense, several significant adjustments and modifications must be performed because the Malcev identities provide a different and more complex structure than those corresponding to Jacobi identities.…”

“…In a second stage, the use of the command elif instead of if (removing some expressions of type fi or od) involves a decrease of the complexity for our algorithm. Taking into consideration the previous comments, we are only explaining those steps in the implementation given in [3], which has been adapted and modified in order to make applicable the implementation to Malcev algebras. In consequence, the description, implementation as well as details of inputs and outputs of Steps 3 to 13 can be found exactly in [3] because there is no difference between considering Lie and Malcev algebras for these steps.…”

“…In previous papers [3,4], the authors developed an algorithmic method, which computes abelian subalgebras and ideals of maximal dimension for a given Lie or Leibniz algebra. Now, we are generalizing that method to the case of Malcev algebras in this paper.…”

“…Both values are invariants of M, being very important for many questions, such as, studying Malcev algebra contractions, derivations, and degenerations. This paper is structured as follows: after reviewing some preliminaries in Section 2, Section 3 provides a step-by-step explanation of the algorithmic method focused on computing abelian subalgebras and ideals of Malcev algebras, emphasizing the novelty in relation with the algorithms introduced in [3,4] and their implementation. Next, Section 4 shows an example from a theoretical viewpoint and also of application.…”

Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics

“…More precisely, recalling the necessary conditions of the double extension structure of Section 2, we compute I, an abelian minimal ideal, which is also a minimal abelian ideal. A function that finds a minimal abelian ideal of B can be derived from an algorithm of [46] that computes all abelian ideals of B: we can choose one of minimal dimension among those.…”