On finite-dimensional absolute-valued algebras satisfying (x^p,x^q,x^r)=0

TL;DR

This paper classifies 4-dimensional absolute-valued algebras satisfying specific polynomial identities, determining the number of isomorphism classes for various parameter sets, and explores the existence of subalgebras within these structures.

Contribution

It provides an exhaustive classification of 4D absolute-valued algebras satisfying (x^p,x^q,x^r)=0 using principal isotopes, detailing the number of isomorphism classes for each parameter set.

Findings

Number of isomorphism classes varies: 2, 3, or infinite.

Specific parameter sets yield exactly 2 or 3 classes.

Existence of 2D subalgebras in all classified algebras.

Abstract

By means of principal isotopes lH(a,b) of the algebra lH [Ra 99] we give an exhaustive and not repetitive description of all 4-dimensional absolute-valued algebras satisfying (x^p, x^q, x^r) = 0 for fixed integers p, q, r \in\{1,2\}. For such an algebras the number N(p,q,r) of isomorphism classes is 2 or 3, or is infinite. Concretely 1. N(1,1,1)=N(1,1,2)=N(1,2,1)=N(2,1,1)=2, 2. N(1,2,2)=N(2,2,1)=3, 3. N(2,1,2)=N(2,2,2)=\infty. Besides, each one of the above algebras contains 2-dimensional subalgebras. However, the problem in dimension 8 is far from being completely solved. In fact, there are 8-dimensional absolute-valued algebras, containing no 4- dimensional subalgebras, satisfying (x^2,x,x^2)=(x^2,x^2,x^2)=0.