Pseudo-Euclidean Alternative Algebras

Abstract

In this paper, we transfer the notion of double extension, introduced by Medina and Revoy for quadratic Lie algebras [8], and extended by Benayadi and Baklouti for pseudo-euclidean Jordan algebras [1, 2], to the case of pseudo-euclidean alternative algebras. We show that every pseudo-euclidean alternative algebra, which is irreducible and neither simple nor nilpotent, is a suitable double extension. Moreover, we introduce the notion of generalized double extension of pseudo-euclidean alternative algebras by the one dimensional alternative algebra with zero product. This leads to an inductive classification of nilpotent pseudo-euclidean alternative algebras. A short review of the basics on alternative algebras and their connections to some other algebraic structures is also provided.