ON SPECIALITY OF BINARY-LIE ALGEBRAS

Abstract

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 [Formula: see text] that cannot be embedded in A-s for an assocyclic superalgebra A. We use the Grassmann envelope of [Formula: see text] 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 [Formula: see text] is isomorphic to the commutator superalgebra [Formula: see text] for a simple binary (-1,1) superalgebra [Formula: see text].