An equivalence in generalized almost-Jordan algebras

• Guzzo Jr., Henrique ^[1] ; Labra, Alicia ^[2]
  1. [1] Universidade de São Paulo

     Universidade de São Paulo

     Brasil

     
  2. [2] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 35, Nº. 4, 2016, págs. 505-519
• Idioma: inglés
• DOI: 10.4067/S0716-09172016000400011
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• Resumen

  • 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-Jordanalgebras. 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.

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