Malcev classification for the variety of left-symmetric algebras

TL;DR

This paper explores the classification of subvarieties within left-symmetric algebras, establishing connections to known algebraic varieties and constructing explicit bases for certain subvarieties.

Contribution

It introduces a classification framework for subvarieties of left-symmetric algebras and constructs explicit bases for specific cases, linking them to alternative, assosymmetric, and Zinbiel algebras.

Findings

Identified three classes of subvarieties within left-symmetric algebras.

Established relations between these subvarieties and well-known algebraic varieties.

Constructed bases for free algebras in certain subvarieties and derived identities for operations.

Abstract

In this paper, we study three classes of subvarieties inside the variety of left-symmetric algebras. We show that these subvarieties are naturally related to some well-known varieties, such as alternative, assosymmetric and Zinbiel algebras. For certain subvarieties of the varieties of alternative and assosymmetric algebras, we explicitly construct bases of the corresponding free algebras. We then define the commutator and anti-commutator operations on these algebras and derive a number of identities satisfied by these operations in all degrees up to $4$.