Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 107, 20 pages      arXiv:2406.10947      https://doi.org/10.3842/SIGMA.2024.107

The Algebraic and Geometric Classification of Compatible Pre-Lie Algebras

Hani Abdelwahab a, Ivan Kaygorodov b and Abdenacer Makhlouf c
a) Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
b) CMA-UBI, University of Beira Interior, Covilhã, Portugal
c) IRIMAS - Département de Mathématiques, University of Haute Alsace, Mulhouse, France

Received August 22, 2024, in final form November 18, 2024; Published online November 28, 2024

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.

Key words: compatible algebra; compatible associative algebra; compatible pre-Lie algebra; algebraic classification; geometric classification.

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