An equivalence in generalized almost-Jordan algebras

Henrique Guzzo Jr., Alicia Labra


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.

Palabras clave

Jordan algebras; generalized almost-Jordan algebras; Lie Triple algebras; baric algebras; álgebras de Jordan; álgebras generalizadas de casi-Jordan; álgebra de Triple Lie; álgebra bárica.

Texto completo:



M. Arenas and A. Labra, On Nilpotency of Generalized Almost-Jordan Right-Nilalgebras, Algebra Colloquium, 15, pp. 69-82, (2008).

M. Arenas, The Wedderburn principal theorem for Generalized AlmostJordan algebras, Comm. in Algebra, 35 (2), pp. 675-688, (2007).

L. Carini, I. R. Hentzel and J. M. Piaccentini-Cattaneo, Degree four Identities not implies by commutativity. Comm. in algebra, 16 (2), pp. 339-357, (1988).

M. Flores and A. Labra, (2015). Representations of Generalized Almost-Jordan Algebras, Comm. in Algebra, 43 (8), pp. 3373-3381, (2015).

I. R. Hentzel and L. A. Peresi, Almost Jordan Rings, Proc. of A.M.S. 104 (2), pp. 343-348, (1988).

I. R. Hentzel and A. Labra, On left nilalgebras of left nilindex four satisfying an identity of degree four. Internat. J. Algebra Comput. 17, (1), pp. 27-35, (2007).

J. M. Osborn, Commutative algebras satisfying an identity of degree four, Proc. A.M.S. 16, pp. 1114-1120, (1965).

J. M. Osborn, Identities of non-associative algebras, Canad. J. Math. 17, pp. 78-92, (1965).

H. Petersson, Zur Theorie der Lie-Tripel-Algebren, Math. Z. 97, pp. 1-15, (1967).

R. Schafer, Introduction on nonassociative algebras, Academic Press, N. York, (1966).

A. V. Sidorov, Lie triple algebras, Translated from Algebra i Logika, 20 (1), pp. 101-108, (1981).


Enlaces refback

  • No hay ningún enlace refback.