The algebraic and geometric classification of commutative post-Lie algebras
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| Format: | Preprint |
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2026
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| _version_ | 1866912866340175872 |
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| author | Abdelwahab, Hani Abdurasulov, Kobiljon Kaygorodov, Ivan |
| author_facet | Abdelwahab, Hani Abdurasulov, Kobiljon Kaygorodov, Ivan |
| contents | 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 $n$-dimensional {\rm CPA}s and gave the algebraic classification of $3$-dimensional {\rm CPA}. We also developed another method for classifying $n$-dimensional nilpotent {\rm CPA}s from nilpotent {\rm CPA}s of smaller dimension and gave the algebraic classification of $4$-dimensional nilpotent {\rm CPA}s. Based on the obtained results, we present the geometric classifications of complex $3$-dimensional and $4$-dimensional nilpotent {\rm CPA}s. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_00614 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The algebraic and geometric classification of commutative post-Lie algebras Abdelwahab, Hani Abdurasulov, Kobiljon Kaygorodov, Ivan Rings and Algebras 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 $n$-dimensional {\rm CPA}s and gave the algebraic classification of $3$-dimensional {\rm CPA}. We also developed another method for classifying $n$-dimensional nilpotent {\rm CPA}s from nilpotent {\rm CPA}s of smaller dimension and gave the algebraic classification of $4$-dimensional nilpotent {\rm CPA}s. Based on the obtained results, we present the geometric classifications of complex $3$-dimensional and $4$-dimensional nilpotent {\rm CPA}s. |
| title | The algebraic and geometric classification of commutative post-Lie algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2602.00614 |