Pilar Benito

Models of the octonions and G2

Different models of the Cayley algebras and of their Lie algebras of derivations are given, based on some distinguished subalgebras of the later ones.

Rota–Baxter Operators on Quadratic Algebras

We prove that all Rota–Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota–Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota–Baxter operators and the solutions to the alternative Yang–Baxter equation on the Cayley–Dickson algebra. We also investigate the Rota–Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.

Commutative n-Ary Leibniz Algebras

We prove that there are no simple commutative n-ary Leibniz algebras of arbitrary dimension over fields of characteristic zero or greater than n. This result extends previous work of Pojidaev and Elduque.

Quadratic 2-step Lie algebras: Computational algorithms and classification

Taking into account the theoretical results and guidelines given inthis work, we introduce a computational method to construct any 2 step nilpotent quadratic algebra of d generators. Along the work we show that the key of the classification of this class of metric algebras relies on certain families of skewsymmetric matrices. Computational examples for d<=8 will be given.

Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

On the set of symmetric matrices over the field , we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these nonassociative structures have the orthogonal Lie algebra as derivation algebra. This gives an embedding for . We obtain a sequence of reductive pairs that provides a family of irreducible Lie-Yamaguti algebras. In this paper, we explain in detail the construction of these Lie-Yamaguti algebras. In the cases , we use computer algebra to determine the polynomial identities of degree ; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.