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## versión impresa ISSN 0716-0917

### Proyecciones (Antofagasta) vol.35 no.4 Antofagasta dic. 2016

#### http://dx.doi.org/10.4067/S0716-09172016000400011

An equivalence in generalized almost-Jordan algebras

Henrique Guzzo Jr.

Brasil

Alicia Labra

Chile

ABSTRACT

In this paper we work with the variety of commutative algebras satisfying the identity β((x2y)x — ((yx)x)x) +γ(x3y — ((yx)x)x) = 0, where β, γ are scalars.    They are called generalized almost-Jordan

algebras. We prove that this variety is equivalent to the variety of commutative algebras satisfying (3β + γ)(Gy(x,z,t) — Gx(y,z,t)) + (β + 3γ)(J(x,z,t)y — J(y,z,t)x) = 0, for all x,y,z,t ∈ A, where J(x,y,z) = (xy)z+(yz)x+(zx)y and Gx(y,z,t) = (yz,x,t)+(yt,x,z)+ (zt,x,y). Moreover, we prove that if A is a commutative algebra, then J (x, z, t)y = J (y, z, t)x, for all x, y, z, t ∈ A, if and only if A is a generalized almost-Jordan algebra for β= 1 and γ = —3, that is, A satisfies the identity (x2y)x + 2((yx)x)x — 3x3y = 0 and we study this identity. We also prove that if A is a commutative algebra, then Gy(x,z,t) = Gx(y,z,t), for all x,y,z,t ∈ A, ifand only if A is an almost-Jordan or a Lie Triple algebra.

Subjclass : Primary 17A30; Secondary 17C50.

Keywords : Jordan algebras, generalized almost-Jordan algebras, Lie Triple algebras, baric algebras.

Thanks : Part if this research was done while the first author was visiting the Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, supported by the FONDECYT grant 1120844 and the second author supported by Programa de Estimulo a la Excelencia Institucional-Universidad de Chile, was visiting the University ofSão Paulo.

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Henrique Guzzo Jr

Universidade de Sao Paulo Instituto de Matemática e Estatística Rua do Matao, 1010 05508-090, Sao Paulo,

Brazil

e-mail : guzzo@ime.usp.br and

Alicia Labra

Departamento de Matemáticas Universidad de Chile Casilla 653, Santiago,

Chile

e-mail : alimat@uchile.cl

Received : September 2016. Accepted : October 2016 Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons