The Algebraic and Geometric Classification of Compatible Pre-Lie Algebras
Hani Abdelwahab, Ivan Kaygorodov, Abdenacer Makhlouf

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.
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Taxonomy
TopicsAdvanced Topics in Algebra
