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Unital algebras of Hom-associative type and surjective or injective twistings. (English) Zbl 1237.17005

In [J. T. Hartwig, D. Larsson and S. D. Silvestrov, J. Algebra 295, No. 2, 314–361 (2006; Zbl 1138.17012)] Hom-Lie-algebras were studied in the context of deformation of Witt and Virasoro algebras. The basic idea was to obtain a generalization of a given type of algebraic structure by adjoining as an additional piece of data linear maps twisting the defining identities. In the search of a counterpart of the associative algebras in the context of Hom-Lie algebras, Hom-associative algebras have been introduced in [A. Makhlouf and S. D. Silvestrov, J. Gen. Lie Theory Appl. 2, No. 2, 51–64 (2008; Zbl 1184.17002)], where it was shown that the commutator bracket of a Hom-associative algebra gives rise to a Hom-Lie algebra.
Hom-associative algebras have been a subject of recent intensive study due to their rich structure theory and the fact that constructions coming from their classical counterparts have been found to transfer to a meaningful extent. Also helpful in this context is the availability of computational tools which greatly facilitate the search for examples and the proof of equational theorems about Hom-associative structures. It was pointed out in [Y. Frégier and A. Gohr, J. Gen. Lie Theory Appl. 4, Article ID G101001, 16 p. (2010; Zbl 1281.17002)] that in the process of defining a twisted notion of, for instance, associativity or the Jacobi identity, there are some choices left on where to apply the twisting. At least in the Hom-associative category it does not seem to be the case that only one of the choices leads to an interesting theory.
In the paper under review the authors introduce a common generalizing framework for alternative types of Hom-associative algebras. They show that the observation that unital Hom-associative algebras with surjective or injective twisting map are already associative has a generalization in this new framework. They also show by construction of a counterexample that another such generalization fails even in a very restricted particular case. Finally, the authors discuss an application of these observations by answering in the negative the question whether nonassociative algebras with unit such as the octonions may be twisted by the composition trick into Hom-associative algebras.

MSC:

17A30 Nonassociative algebras satisfying other identities
16Y99 Generalizations
17A01 General theory of nonassociative rings and algebras
17A20 Flexible algebras
17D25 Lie-admissible algebras
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