ON SPECIALITY OF BINARY-LIE ALGEBRAS

In the present work, binary-Lie, assocyclic, and binary (-1,1) algebras are studied. We prove that, for every assocyclic algebra A, the algebra A(-) is binary-Lie. We find a simple non-Malcev binary-Lie superalgebra T that cannot be embedded in A(-s) for an assocyclic superalgebra A. We use the Grassmann envelope of T to prove the similar result for algebras. This solve negatively a problem by Filippov (see [1, Problem 2.108]). Finally, we prove that the superalgebra T is isomorphic to the commutator superalgebra A(-s) for a simple binary (-1,1) superalgebra A.FAPESP[2007/58048-8]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP[2005/60337-2]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq[305344/2009-9]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq