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Speciality and deformations of algebras. (English) Zbl 1027.17001

Bahturin, Yu. (ed.), Algebra. Proceedings of the international algebraic conference on the occasion of the 90th birthday of A. G. Kurosh, Moscow, Russia, May 25-30, 1998. Berlin: Walter de Gruyter. 345-356 (2000).
From the author’s introduction: The notion of speciality has come from the theory of Jordan algebras. A Jordan algebra is called special if it admits an isomorphic embedding in an associative algebra with respect to a symmetrized multiplication \(a\cdot b= \frac 12 (ab+ba)\). The variety generated by all special algebras neither coincides with the class of all Jordan algebras, nor with the class of all special Jordan algebras. The algebras of this variety are called \(i\)-special. Both speciality and \(i\)-speciality can also be naturally defined for superalgebras.
Here we consider the speciality problem in a more general framework that includes also the problem of embedding Malcev algebras into skew-symmetrized alternative algebras, and that of a linear representability of Akivis algebras. Our main purpose is to show that the methods of deformation theory could be applied to speciality problems.
For the entire collection see [Zbl 0933.00023].

MSC:

17A30 Nonassociative algebras satisfying other identities
17D05 Alternative rings
17D10 Mal’tsev rings and algebras
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