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

TL;DR

This paper investigates certain division algebras over fields of characteristic zero that satisfy specific associativity conditions involving powers of elements, revealing their structural properties and classifications, especially in low dimensions.

Contribution

It characterizes division algebras satisfying particular power-associativity conditions, showing they are quadratic in four dimensions and exploring implications of units and left-units.

Findings

Such algebras are third power-associative if they have a unit.

In degree ≤ 4, these algebras are power-commutative.

Any 4-dimensional real division algebra with a unit satisfying the conditions is quadratic.

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 $\leq 4$ 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".