Division algebras satisfying $(x^p,x^q,x^r)=0$

Abstract

We study algebras $A,$ over a field of characteristic zero, satisfying $(x^p,x^q,x^r)=0$ for $p,q,r$ in $1,2.$ The existence of a unit element in such algebras leads to the third power-associativity. If, in addition, $A$ has degree $\le 4$ then $A$ is power-commutative. We deduce that any 4-dimensional real division algebra, with unit element, satisfying $(x^p,x^q,x^r)=0$ is quadratic. This persists for $(x,x^q,x^r)=0$ if we replace the word "unit" by "left-unit".

References in corpus (1)

• Algebres non associatives normees de division. Classification des algebres reelles de Jordan non commutatives de division lineaire de dimension 8