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Affinization of dendriform $\md$-bialgebras, Lie bialgebras and solutions of classical Yang-Baxter equation

Bo Hou

TL;DR

The work develops an affinization framework for dendriform $D$-bialgebras to construct completed ASI and Lie bialgebras, using tensor products with quadratic $\mathbb{Z}$-graded perm algebras. It provides two parallel routes to the induced Lie algebra on $D\otimes B$—an associative-route via a perm-tensored product and a pre-Lie-route via a pre-Lie algebra—with a proven equivalence and a unifying commutative diagram across dendriform, pre-Lie, ASI and Lie bialgebras. The paper further links Yang–Baxter theory across these settings: symmetric solutions of the pre-Lie YBE yield CYBE solutions in the induced Lie algebra, and triangular/quasi-triangular/factorizable properties transfer through affinization. A central result (Theorem II) characterizes when affinization produces completed ASI bialgebras and shows the converse, connecting operadic dualities with explicit constructions via $\nu_{\omega}$ and completed tensor products. Overall, the results connect dendriform, pre-Lie, ASI and Lie bialgebras through concrete affinization procedures and operator-theoretic insights, enriching the semialgebraic framework for Yang–Baxter equations and their bialgebra realizations.

Abstract

In this paper, we mainly discuss how to use dendriform $\md$-bialgebras to construct Lie bialgebras and the relationship between the solutions of their corresponding Yang-Baxter equations. We provide two methods for obtaining Lie algebras from dendriform algebras using the tensor product with perm algebras, one by means of associative algebras and the other by means of pre-Lie algebras. We elevate both approaches to the level of bialgebras and prove that the Lie bialgebraa obtained using these two approaches are the same. There is a correspondence between symmetric solutions of the dendriform Yang-Baxter equation in dendriform algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the Lie algebras induced from the dendriform algebras. The connections between triangular bialgebra structures, $\mathcal{O}$-operators related to the solutions of these Yang-Baxter equations are discussed in detail. During the discussion, we also present a method for constructing infinite-dimensional antisymmetric infinitesimal bialgebra by using the affineization of dendriform $\md$-bialgebras.

Affinization of dendriform $\md$-bialgebras, Lie bialgebras and solutions of classical Yang-Baxter equation

TL;DR

The work develops an affinization framework for dendriform -bialgebras to construct completed ASI and Lie bialgebras, using tensor products with quadratic -graded perm algebras. It provides two parallel routes to the induced Lie algebra on —an associative-route via a perm-tensored product and a pre-Lie-route via a pre-Lie algebra—with a proven equivalence and a unifying commutative diagram across dendriform, pre-Lie, ASI and Lie bialgebras. The paper further links Yang–Baxter theory across these settings: symmetric solutions of the pre-Lie YBE yield CYBE solutions in the induced Lie algebra, and triangular/quasi-triangular/factorizable properties transfer through affinization. A central result (Theorem II) characterizes when affinization produces completed ASI bialgebras and shows the converse, connecting operadic dualities with explicit constructions via and completed tensor products. Overall, the results connect dendriform, pre-Lie, ASI and Lie bialgebras through concrete affinization procedures and operator-theoretic insights, enriching the semialgebraic framework for Yang–Baxter equations and their bialgebra realizations.

Abstract

In this paper, we mainly discuss how to use dendriform -bialgebras to construct Lie bialgebras and the relationship between the solutions of their corresponding Yang-Baxter equations. We provide two methods for obtaining Lie algebras from dendriform algebras using the tensor product with perm algebras, one by means of associative algebras and the other by means of pre-Lie algebras. We elevate both approaches to the level of bialgebras and prove that the Lie bialgebraa obtained using these two approaches are the same. There is a correspondence between symmetric solutions of the dendriform Yang-Baxter equation in dendriform algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the Lie algebras induced from the dendriform algebras. The connections between triangular bialgebra structures, -operators related to the solutions of these Yang-Baxter equations are discussed in detail. During the discussion, we also present a method for constructing infinite-dimensional antisymmetric infinitesimal bialgebra by using the affineization of dendriform -bialgebras.
Paper Structure (8 sections, 34 theorems, 100 equations)

This paper contains 8 sections, 34 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on pre-Lie algebras and dendriform algebras
  3. Quasi-triangular Lie bialgebras induced by pre-Lie bialgebras
  4. Infinite-dimensional ASI bialgebras vis affinization of dendriform $\mathrm{D}$-bialgebras
  5. Affinization of dendriform algebras and dendriform coalgebras
  6. Completed ASI bialgebras from dendriform $\mathrm{D}$-bialgebra
  7. Triangular ASI bialgebras from triangular dendriform $\mathrm{D}$-bialgebras
  8. Triangular Lie bialgebras vis dendriform $\mathrm{D}$-bialgebras

Key Result

Proposition 2.5

Let $(D, \prec, \succ)$ be a dendriform algebra. Define a bilinear operator $\diamond: D\otimes D\rightarrow D$ by for any $d_{1}, d_{2}\in D$. Then $(D, \diamond)$ is a pre-Lie algebra, called the induced pre-Lie algebra from $(D, \prec, \succ)$.

Theorems & Definitions (74)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: AguBai
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8: ValLZB
  • Proposition 2.9
  • proof
  • ...and 64 more