Nonassociative algebras of biderivation-type

Abstract

The main purpose of this paper is to study the class of Lie-admissible algebras (A,.) such that its product is a biderivation of the Lie algebra (A,[,]), where [,] is the commutator of the algebra (A,.). First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Lie algebras, symmetric Leibniz algebras, Lie-admissible left (or right) Leibniz algebras, Milnor algebras, and LR-algebras. Then, we establish results on the structure of these algebras in the case that the underlying Lie algebras are perfect (in particular, semisimple Lie algebras). In addition, we then study flexible ABD-algebras, showing in particular that these algebras are extensions of Lie algebras in the category of flexible ABD-algebras. Finally, we study left-symmetric ABD-algebras, in particular we are interested in flat pseudo-Euclidean Lie algebras where the associated Levi-Civita products define ABD-algebras on the underlying vector spaces of these Lie algebras. In addition, we obtain an inductive description of all these Lie algebras and their Levi-Civita products (in particular, for all signatures in the case of real Lie algebras).