The algebraic and geometric classification of commutative post-Lie algebras
Abstract
We study commutative post-Lie algebras {\rm CPA}s from an algebraic point of view. Firstly, we find some new identities in {\rm CPA}, which shows that the commutative multiplication gives a medial and derived commutative associative algebra. As corollaries, we have that there are no simple nontrivial commutative post-Lie algebras and that perfect Lie and centrless perfect commutative associative algebras do not admit nontrivial {\rm CPA} structures. The identities of depolarized {\rm CPA}s are defined. Based on the obtained identities, we developed a method for the classification of -dimensional {\rm CPA}s and gave the algebraic classification of -dimensional {\rm CPA}. We also developed another method for classifying -dimensional nilpotent {\rm CPA}s from nilpotent {\rm CPA}s of smaller dimension and gave the algebraic classification of -dimensional nilpotent {\rm CPA}s. Based on the obtained results, we present the geometric classifications of complex -dimensional and -dimensional nilpotent {\rm CPA}s.
Keywords
Cite
@article{arxiv.2602.00614,
title = {The algebraic and geometric classification of commutative post-Lie algebras},
author = {Hani Abdelwahab and Kobiljon Abdurasulov and Ivan Kaygorodov},
journal= {arXiv preprint arXiv:2602.00614},
year = {2026}
}