Left counital Hopf algebra structures on the free commutative Nijenhuis algebras
Left counital Hopf algebra structures on the free commutative Nijenhuis algebras
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自由交换 Nijenhuis 代数上的左共体 Hopf 代数结构
DOI：
10.1007/s11425-017-9489-0
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通讯作者：
郭锂
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作者：
郑上华;郭锂
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，首先证明了左可数双代数上的自由交换Nijenhuis代数（在右可数双代数不成立的意义下）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，在这个意义下，对极也是唯一右侧的。最后，我们应用这个结果证明了自由交换的左共轭双代数上的左共轭双代数是连通的，是分次的，因而是左共轭右对极的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.