Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The algebraic and geometric classification of commutative post-Lie algebras

Hani Abdelwahab, Kobiljon Abdurasulov, Ivan Kaygorodov

TL;DR

The paper analyzes commutative post-Lie algebras (CPAs) from both algebraic and geometric perspectives, showing that the underlying $(P,\cdot)$ is a medial and derived commutative associative algebra and deriving key consequences for simple and perfect Lie structures. It develops a cohomology-based framework using $Z^{2}(A,A)$ and annihilator extensions to classify CPAs in low dimensions, obtaining a complete algebraic classification of 3-dimensional CPAs (Theorem A1) and a detailed 4-dimensional nilpotent CPA classification (Theorem A2). The geometric classification uses the variety ${\mathcal T}_{n}$ of CPAs, its GL$(V)$-action, and degeneration theory to determine irreducible components and rigid algebras, with explicit results for complex 3-dimensional CPAs and 4-dimensional nilpotent CPAs. Overall, the study provides a thorough algebraic and partial geometric portrait of CPAs in low dimensions, integrating cohomology, automorphism actions, and degeneration analysis to map the landscape of CPA structures. The findings illuminate how CPAs sit inside broader nonassociative varieties and offer a roadmap for extending classifications to higher dimensions.

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 $n$-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.

The algebraic and geometric classification of commutative post-Lie algebras

TL;DR

The paper analyzes commutative post-Lie algebras (CPAs) from both algebraic and geometric perspectives, showing that the underlying is a medial and derived commutative associative algebra and deriving key consequences for simple and perfect Lie structures. It develops a cohomology-based framework using and annihilator extensions to classify CPAs in low dimensions, obtaining a complete algebraic classification of 3-dimensional CPAs (Theorem A1) and a detailed 4-dimensional nilpotent CPA classification (Theorem A2). The geometric classification uses the variety of CPAs, its GL-action, and degeneration theory to determine irreducible components and rigid algebras, with explicit results for complex 3-dimensional CPAs and 4-dimensional nilpotent CPAs. Overall, the study provides a thorough algebraic and partial geometric portrait of CPAs in low dimensions, integrating cohomology, automorphism actions, and degeneration analysis to map the landscape of CPA structures. The findings illuminate how CPAs sit inside broader nonassociative varieties and offer a roadmap for extending classifications to higher dimensions.

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.
Paper Structure (14 sections, 17 theorems, 17 equations)

This paper contains 14 sections, 17 theorems, 17 equations.

Table of Contents

  1. Commutative post-Lie algebras
  2. The algebraic classification of CPAs
  3. CPAs
  4. The method of classification of CPAs with a fixed $"\cdot"$ multiplication
  5. $3$-dimensional CPAs (Theorem A1)
  6. Proof of Theorem A1
  7. Nilpotent CPAs
  8. The method of classification of CPAs with nontrivial annihilator
  9. $4$-dimensional nilpotent CPAs (Theorem A2)
  10. Proof of Theorem A2
  11. The geometric classification of CPAs
  12. Preliminaries: degenerations and geometric classification
  13. The geometric classification of $3$-dimensional CPA
  14. The geometric classification of $4$-dimensional nilpotent CPA

Key Result

Proposition 2

A CPA with one trivial multiplication is isomorphic to a commutative associative algebra or a Lie algebra.

Theorems & Definitions (37)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Corollary 4
  • proof
  • Corollary 5
  • proof
  • Lemma 6
  • proof
  • ...and 27 more