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

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.