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Quasi-triangular dual pre-Poisson bialgebras and its connection with Poisson bialgebras

Bo Hou, Ru Li

TL;DR

The paper develops a comprehensive theory of dual pre-Poisson (DPP) bialgebras, introducing quasi-triangular and factorizable structures and linking them to Poisson and perm-algebra frameworks. It establishes a tight correspondence between factorizable DPP bialgebras and quadratic Rota-Baxter DPP algebras of nonzero weight, and shows that the double of any DPP bialgebra is factorizable. A key contribution is a constructive bridge from finite-dimensional Poisson bialgebras to infinite-dimensional completed DPP bialgebras via tensor products with $mathbb{Z}$-graded perm algebras, preserving coboundary/quasi-triangular/triangular properties through the induced DPYBE and PoiYBE relations. The results provide new tools for generating and analyzing DPP bialgebras, with implications for understanding classical Yang-Baxter-type equations in the dual pre-Poisson setting and for expanding the repertoire of completed bialgebra constructions.

Abstract

In this paper, the notions of quasi-triangular and factorizable dual pre-Poisson bialgebras are introduced. A factorizable dual pre-Poisson bialgebra induces a factorization of the underlying dual pre-Poisson algebra, and the double of any dual pre-Poisson bialgebra is factorizable. We introduce the notion of quadratic Rota-Baxter dual pre-Poisson algebras and show that there is a one-to-one correspondence between factorizable dual pre-Poisson bialgebras and quadratic Rota-Baxter Poisson algebras of nonzero weights. Moreover, a method of constructing infinite-dimensional dual pre-Poisson bialgebras using finite-dimensional Poisson bialgebras is given. We prove that there is a completed dual pre-Poisson bialgebra structure the tensor product of a Poisson bialgebra and a quadratic $\bz$-graded perm algebra, and this completed dual pre-Poisson bialgebra structure is coboundary (resp. quasi-triangular, triangular) if the original Poisson bialgebra is coboundary (resp. quasi-triangular, triangular). The induced factorizable finite-dimensional dual pre-Poisson bialgebras are considered.

Quasi-triangular dual pre-Poisson bialgebras and its connection with Poisson bialgebras

TL;DR

The paper develops a comprehensive theory of dual pre-Poisson (DPP) bialgebras, introducing quasi-triangular and factorizable structures and linking them to Poisson and perm-algebra frameworks. It establishes a tight correspondence between factorizable DPP bialgebras and quadratic Rota-Baxter DPP algebras of nonzero weight, and shows that the double of any DPP bialgebra is factorizable. A key contribution is a constructive bridge from finite-dimensional Poisson bialgebras to infinite-dimensional completed DPP bialgebras via tensor products with -graded perm algebras, preserving coboundary/quasi-triangular/triangular properties through the induced DPYBE and PoiYBE relations. The results provide new tools for generating and analyzing DPP bialgebras, with implications for understanding classical Yang-Baxter-type equations in the dual pre-Poisson setting and for expanding the repertoire of completed bialgebra constructions.

Abstract

In this paper, the notions of quasi-triangular and factorizable dual pre-Poisson bialgebras are introduced. A factorizable dual pre-Poisson bialgebra induces a factorization of the underlying dual pre-Poisson algebra, and the double of any dual pre-Poisson bialgebra is factorizable. We introduce the notion of quadratic Rota-Baxter dual pre-Poisson algebras and show that there is a one-to-one correspondence between factorizable dual pre-Poisson bialgebras and quadratic Rota-Baxter Poisson algebras of nonzero weights. Moreover, a method of constructing infinite-dimensional dual pre-Poisson bialgebras using finite-dimensional Poisson bialgebras is given. We prove that there is a completed dual pre-Poisson bialgebra structure the tensor product of a Poisson bialgebra and a quadratic -graded perm algebra, and this completed dual pre-Poisson bialgebra structure is coboundary (resp. quasi-triangular, triangular) if the original Poisson bialgebra is coboundary (resp. quasi-triangular, triangular). The induced factorizable finite-dimensional dual pre-Poisson bialgebras are considered.
Paper Structure (4 sections, 22 theorems, 92 equations)

This paper contains 4 sections, 22 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Dual pre-Poisson algebras and their representations
  3. Quasi-triangular DPP bialgebras and classical Yang-Baxter equation
  4. Infinite-dimensional DPP bialgebras from Poisson bialgebras

Key Result

Proposition 2.4

Let $(A, \circ, \ast)$ be a DPP algebra, $V$ be a vector space and $\mathfrak{l}, \mathfrak{r}, \tilde{\mathfrak{l}}, \tilde{\mathfrak{r}}: A\rightarrow\mathfrak{gl}(V)$ be four linear maps. Define bilinear maps $\tilde{\circ}, \tilde{\ast}: (A\oplus V)\otimes(A\oplus V)\rightarrow A\oplus V$ by for any $a_{1}, a_{2}\in A$ and $v_{1}, v_{2}\in V$. Then $(V, \mathfrak{l}, \mathfrak{r}, \tilde{\mat

Theorems & Definitions (55)

  • Definition 2.1: Agu
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6: Agu
  • Proposition 2.7: Lu
  • proof
  • Definition 3.1
  • ...and 45 more