The Algebraic and Geometric Classification of Compatible Pre-Lie Algebras

TL;DR

This paper develops a method to classify compatible pre-Lie algebras algebraically and geometrically, specifically focusing on complex 2-dimensional cases and related algebraic structures.

Contribution

It introduces a new method for algebraic classification of compatible pre-Lie algebras based on existing classifications, and extends this to geometric classification of algebra varieties.

Findings

Classified complex 2-dimensional compatible pre-Lie algebras.

Classified related algebraic structures like commutative associative and Novikov algebras.

Described irreducible components of algebra varieties.

Abstract

In this paper, we develop a method to obtain the algebraic classification of compatible pre-Lie algebras from the classification of pre-Lie algebras of the same dimension. We use this method to obtain the algebraic classification of complex 2-dimensional compatible pre-Lie algebras. As a byproduct, we obtain the classification of complex 2-dimensional compatible commutative associative, compatible associative and compatible Novikov algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components.