Generalized Jordan algebras

AbstractWe study commutative algebras which are generalizations of Jordan algebras. The associator is defined as usual by (x,y,z) = (xy)z−x(yz). The Jordan identity is (x2,y,x) = 0. In the three generalizations given below, t, β, and γare scalars. ((xx)y)x+t((xx)x)y=0, ((xx)x)(yx)−(((xx)x)y)x=0, β((xx)y)x+γ((xx)x)y−(β+γ)((yx)x)x=0. We show that with the exception of a few values of the parameters, the first implies both the second and the third. The first is equivalent to the combination of ((xx)x)x=0 and the third. We give examples to show that our results are in some reasonable sense, the best possible

Simple (γ, δ) algebras are associative

AbstractA (γ, δ) algebra over a field F is a nonassociative algebra satisfying an identity of the form, (a, b, c) + γ(b, a, c) + δ(c, a, b) = 0, for fixed γ, δ ϵ F, and γ2 − δ2 + δ = 1. We assume that F is of characteristic ≠ 2, ≠ 3; however, we do not assume that the algebra is finite-dimensional over F. We show that any simple (γ, δ) algebra is associative with the possible exception of the cases (± 1, 0) and (1, 1). The approach used in this paper is to represent the identities by matrices by way of the group algebra representation. This enables us to manipulate identities by the well-known techniques of matrix theory

Bernstein algebras given by symmetric bilinear forms

AbstractLet (A, ω) be a finite dimensional Bernstein algebra and N the kernel of ω. We study the algebras where dim N2 is 1. The algebras fall into two general classes. For the first of these classes we give the multiplication tables for the complete set of nonisomorphic algebras. For the second of these classes we give the multiplication tables for what we call “complete algebras.” We show that any algebra of the second class can be embedded in a complete algebra. The multiplication in the complete algebras is easy to describe. The Bernstein algebras of the second class are then characterized as subalgebras of the complete algebras. For Jordan Bernstein algebras satisfying dim N2 = 1 we give the complete classification

Simple (γ, δ) algebras are associative

AbstractA (γ, δ) algebra over a field F is a nonassociative algebra satisfying an identity of the form, (a, b, c) + γ(b, a, c) + δ(c, a, b) = 0, for fixed γ, δ ϵ F, and γ2 − δ2 + δ = 1. We assume that F is of characteristic ≠ 2, ≠ 3; however, we do not assume that the algebra is finite-dimensional over F. We show that any simple (γ, δ) algebra is associative with the possible exception of the cases (± 1, 0) and (1, 1). The approach used in this paper is to represent the identities by matrices by way of the group algebra representation. This enables us to manipulate identities by the well-known techniques of matrix theory

Experimenting with the Identity (xy)z = y(zx)

An experiment with the nonassociative algebra program Albert led to the discovery of the following surprising theorem. Let G be a groupoid satisfying the identity (xy)z = y(zx). Then for products in G involving at least five elements, all factors commute and associate. A corollary is that any semiprime ring satisfying this identity must be commutative and associative, generalizing a known result of Chen.This article is published as Hentzel, Irvin Roy, David P. Jacobs, and Sekhar V. Muddana. "Experimenting with the identity (xy) z= y (zx)." Journal of symbolic computation 16, no. 3 (1993): 289-293. 10.1006/jsco.1993.1047. Posted with permission.</p

On left nilalgebras of left nilindex four satisfying an identity of degree four

We extend the concept of commutative nilalgebras to commutative algebras which are not power associative. We shall study commutative algebras A over fields of characteristic ≠ 2, 3 which satisfy the identities x(x(xx)) = 0 and β{x(y(xx)) - x(x(xy))} + γ{y(x(xx)) - x(x(xy))} = 0. In these algebras the multiplication operator was shown to be nilpotent by Correa, Hentzel and Labra [2]. In this paper we prove that for every x ∈ A we have A(A((xx)(xx))) = 0. We prove that there is an ideal I of A satisfying AI = IA = 0 and A/I is power associative. © World Scientific Publishing Company