An equivalence in generalized almost-Jordan algebras
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...
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| Autores principales: | , |
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| Lenguaje: | English |
| Publicado: |
Universidad Católica del Norte, Departamento de Matemáticas
2016
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| Materias: | |
| Acceso en línea: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172016000400011 |
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| Sumario: | 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. |
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