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Subassociative algebras. (English) Zbl 1014.17002
Summary: An algebra is subassociative if the associator \([x, y, z]\) of any three elements \(x, y, z\) is their linear combination. In this paper we prove that any such algebra is Lie-admissible and that almost any such algebra is proper in the sense that there exists an invariant bilinear form \(A\) for which there holds the following identity: \([x, y, z] = A(y, z)x-A(x,y)z\) which enables a close connection with associative algebras. We discuss also the improper subassociative algebras.
MSC:
17A30 Nonassociative algebras satisfying other identities
17D25 Lie-admissible algebras
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