On Finite-Dimensional Absolute-Valued Algebras Satisfying (x^p, x^q, x^r) = 0

Abstract

By means of principal isotopes ℍ(a, b) of the algebra ℍ [25], 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 ∈ {1, 2}. For such algebras, the number N(p, q, r) of isomorphism classes is either 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) = ∞.

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.

Key Words:

• Absolute-valued algebra
• Central (flexible) idempotent
• Principal isotopes of ℍ

ACKNOWLEDGMENTS

The authors are indebted to the referee for his/her remarks, comments, and suggestions which allowed a clear simplification in the calculations and the improvement in the writing of the article.

The authors are partially supported by the project I+ D MCYT MTM 2007- 65959.

Notes

Communicated by M. Bresar.

To the memory of Nadia Benhoussa, who died prematurely on January 6th, 2010.