José M. Pérez-Izquierdo

An envelope for Malcev algebras

We prove that for every Malcev algebra M there exist an algebra U(M) and a monomorphism ι:M→U(M)− of M into the commutator algebra U(M)− such that the image of M lies into the alternative center of U(M), and U(M) is a universal object with respect to such homomorphisms. The algebra U(M), in general, is not alternative, but it has a basis of Poincare–Birkhoff–Witt type over M and inherits some good properties of universal enveloping algebras of Lie algebras. In particular, the elements of M can be characterized as the primitive elements of the algebra U(M) with respect to the diagonal homomorphism Δ:U(M)→U(M)⊗U(M). An extension of Ado–Iwasawa theorem to Malcev algebras is also proved.

Ternary Derivations of Generalized Cayley–Dickson Algebras

Abstract Ternary derivations, ternary Cayley derivations and ternary automorphisms are computed over fields of characteristic ≠ 2, 3 for the algebras A t obtained by the Cayley–Dickson duplication process. While the derivation algebra of A t stops growing after t = 3, the ternary derivation algebra significantly decreases in the step from the octonions A 3 to the sedenions A 4, revealing the symmetry lost on that stage.

Hopf algebras with triality

In this joint work with G. Benkart and J.M. Perez-Izquierdo, we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows a new construction of the universal enveloping algebras of Malcev algebras. Our work relays on the approach of Grishkov and Zavarnitsine to groups with triality.

Non-Existence of Coassociative Quantized Universal Enveloping Algebras of the Traceless Octonions

The purpose of this brief note is to prove that any coassociative bialgebra deformation of the universal enveloping algebra of the seven dimensional central simple exceptional Malcev algebra over a field of characteristic zero is cocommutative.

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.

Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.