Anticommutative Algebras Satisfying Standard Identities of Degree Four

Abstract

We define an anticommutative Φ-algebra A(D,a) whose multiplication generalizes the concept of a Jacobi bracket in the form (4). It is proved that A(D,a) is a J-algebra and that it satisfies a standard identity of degree four. A subclass ℳ of algebras A(D,a) over Φ which is connected with some class of 3-Lie algebras is distinguished. We establish a criterion of being simple for factor algebras of non-Lie algebras in ℳ, given a 1-dimensional annihilator, and then use it to construct examples of simple infinite-dimensional (of dimension p3-1) non-Lie J-algebras over a field Φ satisfying standard identities of degree 4, if the characteristic p of Φ is zero (for p > 2). Also, the criterion of algebras belonging to is given.