On Trace Forms on a Class of Commutative Algebras Satisfying an Identity of Degree Four

Abstract

In this paper, we deal with commutative algebras A satisfying the identity 2β{(xy)2 - x2y2} + γ{((xy)x)y + ((xy)y)x - (y2x)x - (x2y)y} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel and Piacentini-Cattaneo [2]. We prove that if the algebra A admits an identity element, then A is associative. We also prove that there exist trace forms on A. Finally, we prove that if A has a non-degenerate trace form, then A satisfies the identity ((yx)x)x = y((xx)x), a generalization of right alternativity. Our results require characteristic ≠ 2, 3.