Bijective 1-cocycles and classification of 3-dimensional Left-symmetric Algebras

TL;DR

This paper introduces a new classification method for 3-dimensional left-symmetric algebras using bijective 1-cocycles, simplifying their complex classification process.

Contribution

It establishes a correspondence between left-symmetric algebras and bijective 1-cocycles, providing a systematic classification procedure.

Findings

Classified all 3-dimensional complex left-symmetric algebras

Established a new approach linking algebras to 1-cocycles

Simplified the classification process for nonassociative algebras

Abstract

Left-symmetric algebras have close relations with many important fields in mathematics and mathematical physics. Their classification is very complicated due to the nonassociativity. In this paper, we re-study the correspondence between left-symmetric algebras and the bijective 1-cocycles. Then a procedure is provided to classify left-symmetric algebras in terms of classification of equivalent classes of bijective 1-cocycles. As an example, the 3-dimensional complex left-symmetric algebras are classified.