自由交换Nijenhuis代数上的左余单位Hopf代数结构  

Left counital Hopf algebra structures on free commutative Nijenhuis algebras

作　　者：郑上华 郭锂 Shanghua Zheng;Li Guo

机构地区：[1]江西师范大学数学与信息科学学院,南昌330022 [2]Department of Mathematics and Computer Science,Rutgers University,Newark,NJ 07102,USA

出　　处：《中国科学：数学》2020年第6期829-846,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11601199和11771190);江西省教育厅科技研究(批准号:GJJ160336);国家留学基金(批准号:201708360057)资助项目。

摘　　要：本文基于自由交换Rota-Baxter代数上的Hopf代数结构,探讨自由交换Nijenhuis代数上的Hopf代数相关结构;借助于上闭链(cocycle)条件证明左余单位双代数(即不满足右余单位性)上的自由交换Nijenhuis代数具有左余单位双代数结构.本文获得更具一般性的结论,连通分次左余单位双代数是左余单位右对极Hopf代数,即其对极只是右侧的.由此证明连通左余单位双代数上的自由交换Nijenhuis代数是连通且分次的,从而,它是左余单位右对极Hopf代数.Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra(in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally apply this result to show that the left counital bialgebra on a free commutative Nijenhuis algebra on a connected left counital bialgebra is connected and graded, hence is a left counital right antipode Hopf algebra.

关 键 词：Nijenhuis代数 双代数 左余单位双代数 连通双代数 左余单位右对极Hopf代数 

分 类 号：O15[理学—数学]