On a remarkable class of left-symmetric algebras and its relationship with the class of Novikov algebras

TL;DR

This paper explores a special class of left-symmetric algebras characterized by a specific identity, examining their properties and connections to Novikov algebras and derivation algebras.

Contribution

It provides new characterizations of these left-symmetric algebras and clarifies their relationship with Novikov and derivation algebras.

Findings

Characterization of a class of left-symmetric algebras satisfying [x,y].z=0

Relationship established between these algebras and Novikov algebras

Insights into affine actions of Lie groups with translation properties

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.