On finite-dimensional absolute-valued algebras satisfying (xᵖ,xq,xʳ)=0

الأصل · EN

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 (xᵖ, xq, xʳ) = 0 for fixed integers p, q, r ∈{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)=∞. 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 (x²,x,x²)=(x²,x²,x²)=0.