Weighted Infinitesimal Bialgebras

DOI：10.16205/j.cnki.cama.2022.0011

 作者 单位 张毅 南京信息工程大学数学与统计学院, 南京 210044.南京信息工程大学江苏省应用数学中心.江苏省系统建模与数据分析国际合作联合实验室 高兴 兰州大学数学与统计学院, 兰州 730000

作为非齐次结合经典Yang-Baxter 方程的代数抽象,带权无穷小双代数在数学和数学物理领域扮演着重要的角色. 本文引入了带权无穷小Hopf模的概念,证明了带权拟三角无穷小单位双代数上的任意模都有一个自然的带权无穷小单位Hopf模结构.利用一种新的方式装饰平面根森林, 并证明根森林的空间,连同它上边的余乘和一组嫁接算子是集合上权为零的自由多重1-余圈无穷小单位双代数. 给出了余乘的一个组合解释.作为应用, 得到了未装饰的平面根森林上的余圈无穷小单位双代数范畴中的初始对象,它也是(非交换)Connes-Kreimer-Hopf代数中的研究对象. 最后,分别从任意带权无穷小双代数和带权交换无穷小双代数导出了两个预李代数,其中第二个构造推广了Novikov 代数上的Gelfand-Dorfman定理.

As an algebraic meaning of the nonhomogenous associative Yang-Baxter equation, weighted infinitesimal bialgebras play an important role in mathematics and mathematical physics. In this paper, the authors introduce the concept of weighted infinitesimal Hopf modules and show that any module carries a natural structure of weighted infinitesimal unitary Hopf module over a weighted quasitriangular infinitesimal unitary bialgebra. They decorate planar rooted forests in a new way, and prove that the space of rooted forests,together with a coproduct and a family of grafting operations, is the free ?-cocycle in-finitesimal unitary bialgebra of weight zero on a set. A combinatorial description of the coproduct is given. As applications, the authors obtain the initial object in the category of cocycle infinitesimal unitary bialgebras on undecorated planar rooted forests, which is the object studied in the (noncommutative) Connes-Kreimer Hopf algebra. Finally, they derive two pre-Lie algebras from an arbitrary weighted infinitesimal bialgebra and weighted commutative infinitesimal bialgebra, respectively. The second construction generalizes the Gelfand-Dorfman theorem on Novikov algebras.