In this paper we provide some new tools for the study of finite-dimensional absolute-valued algebras. We introduce homotopy notions in this field and develop some of their applications. Next, we parametrize these algebras by spin groups and study their isomorphisms. Finally, we introduce a duplication process for the construction of absolute-valued algebras.
We study third-power associative division algebras A over a field K of characteristic different from 2 Those algebras having dimension ≤ 2 are commutative. When K is the field R of real numbers, those algebras having dimension 4 are power-commutative in each of the following two cases:(i) A contains a central element; (ii) A satisfies the additional identity x x 3 x = 0
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