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About relationship between generalized structurable algebras and Lie related triples. (English) Zbl 1164.17005
Editorial comment: The paper is a verbatim copy of the first three sections of K. Noriaki, Adv. Appl. Clifford Algebr. 5, No. 2, 127–140 (1995; Zbl 0854.17002)].
The author considers generalized structurable algebras (i.e., nonassociative algebras over a ring of scalars $$k$$ with a nontrivial derivation $$D$$ introduced in N. Kamiya [Adv. Appl. Clifford Algebr. 5, No. 2, 127–140 (1995; Zbl 0854.17002)]) without 2 and 3-torsion. The following construction is provided. Let $$A$$ be a nonassociative algebra with involution over the ring $$k$$, $$B=A_{12}\oplus A_{23}\oplus A_{31}$$ the sum of copies of the algebra $$A$$ with bracket operation $$[a_{ij},b_{jk}]=ab_{ik}$$ and all other brackets zero. Then by means of Lie related triples the algebra $$B$$ may be furnished with a suitable derivation to become a generalized structurable algebra. Moreover, the vector space sum of the algebra $$B$$ and its inner derivations may also be furnished with suitable product and derivation to become a generalized structurable algebra.
##### MSC:
 17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras) 17B99 Lie algebras and Lie superalgebras 16W25 Derivations, actions of Lie algebras
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