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From pre-Lie bialgebras to phase spaces of Lie algebras: a categorical correspondence

Qi Wang, Xueyi Wang, Jiefeng Liu

TL;DR

The work builds a comprehensive categorical framework linking phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical $\mathfrak s$-matrices, and relative Rota-Baxter operators through coherent homomorphisms and endo/polarization techniques. By translating structural correspondences into functors and equivalences, it unifies endo pre-Lie, endo phase spaces, and endo Manin triples with matched pairs and bialgebras, yielding parallel category structures across the classical objects. It also develops a cohomology theory for symmetric $\mathfrak s$-matrices and a deformation theory via the associated relative Rota-Baxter operators, including a $\psi$-CYBE variant, and establishes precise links between $\mathfrak s$-matrices and RB-operators. The results provide a cohesive, category-theoretic lens on phase spaces, Yang-Baxter-type equations, and their cohomological deformations, with potential applications to integrable systems and geometric structures on Lie groups.

Abstract

This paper establishes a categorical framework for phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical s-matrices, and relative Rota-Baxter operators by introducing the concept of coherent homomorphisms. Starting with endo pre-Lie algebras (pre-Lie algebras equipped with endomorphisms), we extend classical constructions to this enhanced setting, which leads to the notion of coherent endomorphisms for each class of structures. Through polarization, these endomorphisms naturally generalize to coherent homomorphisms, establishing well-defined categories of these algebraic objects. Furthermore, mappings between categories are elevated to functors or equivalences, formalizing interconnections among the constructions. Finally, exploiting the categorical correspondence between s-matrices and relative Rota-Baxter operators, we develop cohomology and deformation of s-matrices, thereby bridging algebraic structures with category-theoretic methods.

From pre-Lie bialgebras to phase spaces of Lie algebras: a categorical correspondence

TL;DR

The work builds a comprehensive categorical framework linking phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical -matrices, and relative Rota-Baxter operators through coherent homomorphisms and endo/polarization techniques. By translating structural correspondences into functors and equivalences, it unifies endo pre-Lie, endo phase spaces, and endo Manin triples with matched pairs and bialgebras, yielding parallel category structures across the classical objects. It also develops a cohomology theory for symmetric -matrices and a deformation theory via the associated relative Rota-Baxter operators, including a -CYBE variant, and establishes precise links between -matrices and RB-operators. The results provide a cohesive, category-theoretic lens on phase spaces, Yang-Baxter-type equations, and their cohomological deformations, with potential applications to integrable systems and geometric structures on Lie groups.

Abstract

This paper establishes a categorical framework for phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical s-matrices, and relative Rota-Baxter operators by introducing the concept of coherent homomorphisms. Starting with endo pre-Lie algebras (pre-Lie algebras equipped with endomorphisms), we extend classical constructions to this enhanced setting, which leads to the notion of coherent endomorphisms for each class of structures. Through polarization, these endomorphisms naturally generalize to coherent homomorphisms, establishing well-defined categories of these algebraic objects. Furthermore, mappings between categories are elevated to functors or equivalences, formalizing interconnections among the constructions. Finally, exploiting the categorical correspondence between s-matrices and relative Rota-Baxter operators, we develop cohomology and deformation of s-matrices, thereby bridging algebraic structures with category-theoretic methods.
Paper Structure (20 sections, 55 theorems, 113 equations)

This paper contains 20 sections, 55 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Phase spaces of Lie algebras, pre-Lie bialgebras and the related structures
  3. Morphisms of bialgebras, Manin triples and matched pairs
  4. Morphism of phase spaces, classical $\mathfrak s$-matrices and relative Rota-Baxter operators
  5. Outline of the paper
  6. Preliminaries
  7. Pre-Lie algebras, representations and cohomologies
  8. Relative Rota-Baxter operators on Lie algebras and their Maurer-Cartan characterizations
  9. Endo pre-Lie algebras, Manin triples and endo symplectic Lie algebras
  10. Endo pre-Lie algebras and their representations
  11. Matched pairs of endo pre-Lie algebras
  12. Endo symplectic Lie algebras and Manin triples of endo pre-Lie algebras
  13. Endo pre-Lie bialgebras
  14. Coherent homomorphisms of pre-Lie bialgebras, Manin triples and phase spaces of Lie algebras
  15. Coherent homomorphisms of phase spaces and Manin triples for pre-Lie algebras
  16. ...and 5 more sections

Key Result

Theorem 2.5

(Bai-1) Let $T:V\to \mathfrak g$ be a relative Rota-Baxter operator on a Lie algebra $(\mathfrak g,[\cdot,\cdot]_\mathfrak g)$ with respect to a representation $(V;\rho)$. Define a multiplication $\cdot^T$ on $V$ by Then $(V,\cdot^T)$ is a pre-Lie algebra.

Theorems & Definitions (120)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Lemma 2.8
  • Proposition 2.9
  • Definition 3.1
  • ...and 110 more