An algorithm for the classification of 3-dimensional complex Leibniz algebras

(Pseudo)digraphs and Leibniz algebra isomorphisms

“…Second, we give a complete list with all nonisomorphic combinatorial structures of 3 vertices associated with each isomorphism class of Leibniz algebras, indicating the structure of each algebra. Then, we compare the results obtained in these sections with the current classifications of 2‐ and 3‐dimensional Leibniz algebras (see Casas et al and Curvier). Finally, we introduce and implement the algorithm that automates all the computations related to the link between combinatorial structures and Leibniz algebras in order to decide if a given combinatorial structure is associated or not with a Leibniz algebra.…”

“…Now, we clarify how all the isomorphism families obtained in Sections 4 and 5 correspond to the well-known classification of 3-dimensional Leibniz algebras. We will focus on non-Lie Leibniz algebras, since the study for 3-dimensional Lie algebras was done in Cáceres et al 9 Table 4 shows all the 3-dimensional non-Lie Leibniz algebras obtained in this paper and in Casas et al 10…”

A study of n-Lie-isoclinic Leibniz algebras

“…Remark 3.14 If g 1 and g 2 are n-Lie-isoclinic Leibniz algebras, then there exists the two-sided ideal J satisfying the requirements of Corollary 3.13 thanks to Corollary 3.12. Other broad class of algebras satisfying the requirements of Corollary 3.13 are Lie-nilpotent Leibniz algebras of class n. Another example of non Lie-nilpotent Leibniz algebra satisfying the requirements of Corollary 3.13 is the three-dimensional Leibniz algebra g 1 with basis {a 1 , a 2 , a 3 }, with bracket operation [a 1 , a 3 ] = a 1 (see algebra 2 d) in the classification given in [6]). Take the two-sided ideal J = {a 2 } , γ Lie n+1 (g 1 ) = {a 1 } , hence the intersection is zero.…”

Algorithmic method to obtain combinatorial structures associated with Leibniz algebras

“…11, as we did in Remark 13. These classes are contained in the classification of 3-dimensional Leibniz algebras given in [5]. At this respect, the list of pseudodigraphs given in Proposition 14 does not correspond to a list of isomorphism classes for Leibniz algebras, but a list of configurations covering all the classes associated with pseudodigraphs.…”