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Noncommutative pre-Poisson bialgebras and relative Rota-Baxter operators

Hongliang Li, Qinxiu Sun

TL;DR

The paper develops a bialgebra theory for coherent noncommutative pre-Poisson algebras, linking coboundary constructions to the noncommutative pre-Poisson Yang-Baxter equation (NPP-YBE) and examining quasi-triangular and factorizable structures. It formulates representations, phase spaces, and Manin triples in this noncommutative setting, and extends the theory through dendriform and pre-Lie coalgebras to coboundary and O-operator viewpoints. A central contribution is establishing a tight bridge between symmetric (or invariant) solutions of NPP-YBE and coboundary bialgebras, as well as showing how O-operators realize these solutions. The results give a robust framework connecting quadratic Rota-Baxter theories with factorizable bialgebras, enabling systematic construction and classification of quasi-triangular and triangular instances with potential applications in noncommutative geometry and integrable systems.

Abstract

In this paper, we develop the bialgebra theory for coherent noncommutative pre-Poisson algebras and establish equivalences among matched pairs, Manin triples, the phase space of noncommutative Poisson algebras and noncommutative pre-Poisson bialgebras. The investigation of coboundary noncommutative pre-Poisson bialgebras naturally leads to the noncommutative pre-Poisson Yang-Baxter equation (NPP-YBE). We prove that a symmetric solution of the NPP-YBE gives rise to a (coboundary) noncommutative pre-Poisson bialgebra. Moreover, we demonstrate how solutions without the symmetry condition can also generate such bialgebras. This motivates the introduction of quasi-triangular and factorizable noncommutative pre-Poisson bialgebras.In particular, we show that a solution of the NPP-YBE with an invariant skew-symmetric part yields a quasi-triangular noncommutative pre-Poisson bialgebra.Such solutions are further interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter noncommutative pre-Poisson algebras and factorizable noncommutative pre-Poisson bialgebras.

Noncommutative pre-Poisson bialgebras and relative Rota-Baxter operators

TL;DR

The paper develops a bialgebra theory for coherent noncommutative pre-Poisson algebras, linking coboundary constructions to the noncommutative pre-Poisson Yang-Baxter equation (NPP-YBE) and examining quasi-triangular and factorizable structures. It formulates representations, phase spaces, and Manin triples in this noncommutative setting, and extends the theory through dendriform and pre-Lie coalgebras to coboundary and O-operator viewpoints. A central contribution is establishing a tight bridge between symmetric (or invariant) solutions of NPP-YBE and coboundary bialgebras, as well as showing how O-operators realize these solutions. The results give a robust framework connecting quadratic Rota-Baxter theories with factorizable bialgebras, enabling systematic construction and classification of quasi-triangular and triangular instances with potential applications in noncommutative geometry and integrable systems.

Abstract

In this paper, we develop the bialgebra theory for coherent noncommutative pre-Poisson algebras and establish equivalences among matched pairs, Manin triples, the phase space of noncommutative Poisson algebras and noncommutative pre-Poisson bialgebras. The investigation of coboundary noncommutative pre-Poisson bialgebras naturally leads to the noncommutative pre-Poisson Yang-Baxter equation (NPP-YBE). We prove that a symmetric solution of the NPP-YBE gives rise to a (coboundary) noncommutative pre-Poisson bialgebra. Moreover, we demonstrate how solutions without the symmetry condition can also generate such bialgebras. This motivates the introduction of quasi-triangular and factorizable noncommutative pre-Poisson bialgebras.In particular, we show that a solution of the NPP-YBE with an invariant skew-symmetric part yields a quasi-triangular noncommutative pre-Poisson bialgebra.Such solutions are further interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter noncommutative pre-Poisson algebras and factorizable noncommutative pre-Poisson bialgebras.
Paper Structure (17 sections, 44 theorems, 107 equations)

This paper contains 17 sections, 44 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. (Noncommutative) Poisson algebras and (noncommutative) pre-Poisson algebras
  3. Quasi-triangular bialgebras
  4. Outline of the paper
  5. (Quasi-) representations and phase spaces of noncommutative Poisson algebras
  6. (Quasi-) representations of noncommutative pre-Poisson algebras
  7. Phase spaces of noncommutative Poisson algebras
  8. Noncommutative pre-Poisson bialgebras and the noncommutative pre-Poisson Yang-Baxter equation
  9. Dendriform bialgebras and pre-Lie bialgebras
  10. Noncommutative pre-Poisson bialgebras
  11. Coboundary noncommutative pre-Poisson bialgebras and the noncommutative pre-Poisson Yang-Baxter equation
  12. Quasi-triangular noncommutative pre-Poisson bialgebras and factorizable noncommutative pre-Poisson bialgebras
  13. Quasi-triangular noncommutative pre-Poisson bialgebras
  14. Factorizable noncommutative pre-Poisson bialgebras
  15. Relative Rota-Baxter operators and quadratic Rota-Baxter noncommutative pre-Poisson algebras
  16. ...and 2 more sections

Key Result

Proposition 2.6

Let $(A,\succ,\prec,\ast)$ be a noncommutative pre-Poisson algebra.

Theorems & Definitions (105)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • Proposition 2.8
  • ...and 95 more