Left counital Hopf algebra structures on free commutative Nijenhuis algebras
Left counital Hopf algebra structures on free commutative Nijenhuis algebras
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DOI：
10.1360/scm-2017-0662
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发表时间：
2017-11
影响因子：
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通讯作者：
Shanghua Zheng;Li Guo
中科院分区：
文献类型：
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作者：
Shanghua Zheng;Li Guo
受自由交换Rota-Baxter代数上的Hopf代数结构的启发，我们研究了自由交换Nijenhuis代数上的Hopf代数相关结构.利用一个上循环条件，证明了左可数双代数上的自由交换Nijenhuis代数（在右可数不成立的意义下）可以扩充为左可数双代数.然后，我们建立了一个一般结果：一个连通的分次左可数双代数是一个左可数右对极Hopf代数，在这个意义上，对极也仅是右侧的.最后，我们应用这个结果证明了自由交换Nijenhuis代数上的左共轭双代数是连通的，是分次的，因此是左共轭右对极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.