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Rota-Type Operators on 2-Dimensional Pre-Lie Algebras

Imed Basdouri, Bouzid Mosbahi, Ahmed Zahari

TL;DR

This work addresses the classification of Rota-Baxter, Reynolds, Nijenhuis, and Averaging operators on all $2$-dimensional complex pre-Lie algebras. By leveraging the known classification of $2$-dimensional pre-Lie algebras and computational tools, the authors derive explicit operator forms for each algebra in the eight isomorphism classes $A_1$–$A_8$, across weights $0$ and $1$. The main results provide detailed operator matrices with precise parameter restrictions, offering a concrete catalog of Rota-type operators in these small non-associative settings. The findings advance understanding of how these operators interact with pre-Lie structure and furnish a resource for constructing related algebraic frameworks, with potential geometric and physical applications.

Abstract

This paper studies Rota-Baxter, Reynolds, Nijenhuis, and Averaging operators on 2-dimensional pre-Lie algebras over $\mathbb{C}$. Using the classification of 2-dimensional pre-Lie algebras and computational tools like Mathematica or Maple, we describe these Rota-type operators in detail. Our results provide a deeper understanding of these operators and their roles in algebraic structures.

Rota-Type Operators on 2-Dimensional Pre-Lie Algebras

TL;DR

This work addresses the classification of Rota-Baxter, Reynolds, Nijenhuis, and Averaging operators on all -dimensional complex pre-Lie algebras. By leveraging the known classification of -dimensional pre-Lie algebras and computational tools, the authors derive explicit operator forms for each algebra in the eight isomorphism classes –, across weights and . The main results provide detailed operator matrices with precise parameter restrictions, offering a concrete catalog of Rota-type operators in these small non-associative settings. The findings advance understanding of how these operators interact with pre-Lie structure and furnish a resource for constructing related algebraic frameworks, with potential geometric and physical applications.

Abstract

This paper studies Rota-Baxter, Reynolds, Nijenhuis, and Averaging operators on 2-dimensional pre-Lie algebras over . Using the classification of 2-dimensional pre-Lie algebras and computational tools like Mathematica or Maple, we describe these Rota-type operators in detail. Our results provide a deeper understanding of these operators and their roles in algebraic structures.
Paper Structure (6 sections, 6 theorems, 38 equations)

This paper contains 6 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Rota-Baxter operator
  4. Reynolds operator
  5. Nijenhuis Operator
  6. Averaging Operator

Key Result

Theorem 1.1

Let $A$ be a nonzero 2-dimensional pre-Lie algebra. Then $A$ is isomorphic to one and only one of the following algebras:

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • ...and 1 more