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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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