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

TL;DR

This paper explores how dendriform ialgebras can be used to construct Lie bialgebras and solutions to the classical Yang-Baxter equation, revealing new links between algebraic structures and integrable systems.

Contribution

It introduces two methods to derive Lie bialgebras from dendriform algebras and establishes their equivalence, along with a novel approach to infinite-dimensional antisymmetric infinitesimal bialgebras.

Findings

Equivalent Lie bialgebras from two dendriform methods

Correspondence between dendriform Yang-Baxter solutions and classical Yang-Baxter solutions

Construction of infinite-dimensional antisymmetric infinitesimal bialgebras

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.