Associo-symmetric algebras

Abstract:

Let A be an algebra over a field F satisfying $(x,x,x) = 0$ with a function $g:A \times A \times A \to F$ such that $(xy)z = g(x,y,z)x(yz)$ for all x, y, z in A. If $g({x_1},{x_2},{x_3}) = g({x_{1\pi }},{x_{2\pi }},{x_{3\pi }})$ for all $\pi$ in ${S_3}$ and all ${x_1},{x_2},{x_3}$ in A then A is called an associo-symmetric algebra. It is shown that a simple associo-symmetric algebra of degree $> 2$ or degree $= 1$ over a field of characteristic $\ne 2$ is associative. In addition a finite-dimensional semisimple algebra in this class has an identity and is a direct sum of simple algebras.