On a Remarkable Class of Left-symmetric Algebras and its Relationship with the Class of Novikov Algebras

Abstract

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G, G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x, y]·z = 0. We derive some basic characterizations of such left-symmetric algebras, and we highlight their relationships with the so-called Novikov algebras and derivation algebras.