Left-symmetric algebras of derivations of free algebras

TL;DR

This paper introduces a left-symmetric algebra structure on derivations of free algebras, explores nilpotent elements, and classifies simple and Novikov derivation algebras, advancing understanding of their algebraic properties.

Contribution

It defines a new left-symmetric algebra structure on derivations of free algebras and classifies simple and Novikov derivation algebras, providing new insights into their structure.

Findings

The commutator algebra of the introduced structure is the Lie algebra of derivations.

Characterization of left and right nilpotent elements in these algebras.

Description of simple and Novikov derivation algebras and generation properties.

Abstract

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric $m$-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.