Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Averaging pre-Lie bialgebras

Lin Gao, Mengke Yang, Yuanyuan Zhang

TL;DR

The paper develops a comprehensive framework for averaging pre-Lie algebras by introducing representation theory, matched pairs, and Manin triples, and by establishing their equivalence with averaging pre-Lie bialgebras under suitable conditions. It then builds averaging pre-Lie bialgebras through averaging operators on quadratic Rota-Baxter pre-Lie algebras and via duality, connecting these structures to symmetric solutions of the admissible classical Yang-Baxter equation (CYBE) through relative Rota-Baxter operators. Dualities and semi-direct constructions are leveraged to relate averaging pre-Lie bialgebras to their Leibniz and Lie counterparts, culminating in a pathway from averaging pre-Lie to averaging Lie bialgebras. Overall, the work integrates operators, r-matrices, and bialgebra concepts to advance the theory of averaging structures in the pre-Lie and Lie settings, with potential implications for related algebraic and mathematical-physics applications.

Abstract

In this paper, we first introduce representations of averaging pre-Lie algebras and study their matched pairs, Manin triples, and bialgebra theories. We prove that these three notions are equivalent under certain conditions. Moreover, by introducing averaging operators on quadratic Rota-Baxter pre-Lie algebras, we show that such operators give rise to averaging pre-Lie bialgebras. Then we introduce the notion of admissible classical Yang-Baxter equations in averaging pre-Lie algebras, as well as the relative Rota-Baxter operators on averaging pre-Lie algebras, and show that the relative Rota-Baxter operators on averaging pre-Lie algebras yield symmetric solutions of admissible classical Yang-Baxter equations in averaging pre-Lie algebras. Finally, we show that every averaging pre-Lie bialgebra induces an averaging Lie bialgebra.

Averaging pre-Lie bialgebras

TL;DR

The paper develops a comprehensive framework for averaging pre-Lie algebras by introducing representation theory, matched pairs, and Manin triples, and by establishing their equivalence with averaging pre-Lie bialgebras under suitable conditions. It then builds averaging pre-Lie bialgebras through averaging operators on quadratic Rota-Baxter pre-Lie algebras and via duality, connecting these structures to symmetric solutions of the admissible classical Yang-Baxter equation (CYBE) through relative Rota-Baxter operators. Dualities and semi-direct constructions are leveraged to relate averaging pre-Lie bialgebras to their Leibniz and Lie counterparts, culminating in a pathway from averaging pre-Lie to averaging Lie bialgebras. Overall, the work integrates operators, r-matrices, and bialgebra concepts to advance the theory of averaging structures in the pre-Lie and Lie settings, with potential implications for related algebraic and mathematical-physics applications.

Abstract

In this paper, we first introduce representations of averaging pre-Lie algebras and study their matched pairs, Manin triples, and bialgebra theories. We prove that these three notions are equivalent under certain conditions. Moreover, by introducing averaging operators on quadratic Rota-Baxter pre-Lie algebras, we show that such operators give rise to averaging pre-Lie bialgebras. Then we introduce the notion of admissible classical Yang-Baxter equations in averaging pre-Lie algebras, as well as the relative Rota-Baxter operators on averaging pre-Lie algebras, and show that the relative Rota-Baxter operators on averaging pre-Lie algebras yield symmetric solutions of admissible classical Yang-Baxter equations in averaging pre-Lie algebras. Finally, we show that every averaging pre-Lie bialgebra induces an averaging Lie bialgebra.
Paper Structure (12 sections, 36 theorems, 118 equations, 1 figure, 2 tables)

This paper contains 12 sections, 36 theorems, 118 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Averaging pre-Lie algebras
  3. Averaging pre-Lie algebras and its representations
  4. Matched pair of averaging pre-Lie algebras
  5. Manin triple of averaging pre-Lie algebras
  6. Averaging pre-Lie bialgebras and averaging operators on quadratic Rota-Baxter pre-Lie algebras
  7. Averaging pre-Lie bialgebras
  8. Averaging operators on quadratic Rota-Baxter pre-Lie algebras
  9. Admissible classical Yang-Baxter equation in the averaging pre-Lie algebra
  10. Admissible classical Yang-Baxter equation in the averaging pre-Lie algebra
  11. Relative Rota-Baxter operator on averaging pre-Lie algebras
  12. Balanced averaging pre-Lie bialgebras

Key Result

Proposition 2.5

Let $\bigl((A,\circ),P\bigr)$ be an averaging pre-Lie algebra. Define a bilinear operation Then $(A,[\cdot ,\cdot ]_P)$ is a Leibniz algebra, i.e. The Leibniz algebra $(A,[\cdot ,\cdot ]_P)$ is called the induced Leibniz algebra of the averaging pre-Lie algebra, denoted by $A_P$.

Figures (1)

  • Figure :

Theorems & Definitions (102)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 92 more