Duplication of algebras. IV (Q648949)

This is posthume continuation of a series of papers by the author. Given a \(K\) algebra \(A\) we denote bt \(D_{K,nc}(A)\) the algebra structure defined on the \(K\)-vector space \(A\otimes A\) by the product \((x\otimes y)(x'\otimes y')=xy\otimes x'y'\). This is called the non-commutative duplicated of \(A\). If \(A\) is commutative, its duplicated is not commutative in general (the same holds for associativity). Nevertheless, if \(A\) is a finite dimensional associative algebra, its duplicated is 4-associative. This paper gives conditions in which the duplicated algebra of a given algebra \(A\) is flexible, Lie admissible or \(n\)-associative. At the end of the paper a short note wirtten by A. Micali, M. Ouattara and N. B. Pilabré explains how this paper was prepared after the death of Prof. Arie Hendrik Boers.