带权无穷小单位双代数和带权结合杨巴方程

Weighted Infinitesimal Unitary Bialgebras and Weighted Associative Yang-Baxter Equations

本文通过构造一个合适的余乘,在一类含单位代数上构造了带权无穷小单位双代数.进一步,构造了Aguiar观点下的无穷小单位Hopf代数.作为应用,在含单位代数上分别构造了一个预李代数和一个新的李代数结构.接着,引入了带权拟三角无穷小单位双代数的定义,推广了Aguiar介绍的拟三角无穷小双代数.然后,证明了任一带权拟三角无穷小单位双代数都有一个叶型代数结构.最后,构造了矩阵代数上权为一λ的结合杨巴方程的解和权为λ的罗巴算子之间的双射.

In this paper,we equip a unitary algebra with a weighted infinitesimal unitary bialgebraic structure via a construction of a suitable coproduct.Furthermore,an infinitesimal unitary Hopf algebra,under the view of Aguiar,is constructed on a unitary algebra.As an application,we construct a pre-Lie algebraic structure and then a new Lie algebraic structure on a unitary algebra.We introduce the concept of weighted quasitriangular infinitesimal unitary bialgebras,which generalize the quasitriangular infinitesimal bialgebras initiated by Aguiar.We then show that any weighted quasitriangular infinitesimal unitary bialgebra has a dendriform algebraic structure.Finally,we give a bijection between the solutions of the associative YangBaxter equation of weight-λand Rota-Baxter operators of weightλon matrix algebras.