λ-TD algebras, generalized shuffle products and left counital Hopf algebras
λ-TD algebras, generalized shuffle products and left counital Hopf algebras
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DOI：
10.1142/s0219498824500993
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发表时间：
2022-07
影响因子：
0.8
通讯作者：
Hengyi Luo;Shanghua Zheng
中科院分区：
文献类型：
--
作者：
Hengyi Luo;Shanghua Zheng
一个bstract。运算代数理论在数学和物理中起着举足轻重的作用。在这篇文章中，我们引入了一个λ-TD代数，它适当地包括了Rota-Baxter代数和TD-代数。交换代数上自由交换λ-TD代数的显式构造是由广义Shuffle积得到的，称为λ-TD Shuffle积。在适当的1-上循环条件下，证明了自由交换λ-TD代数具有左对数双代数结构。进一步地，每个连通fi双代数都是Hopf代数的经典结果被推广到左数双代数的情形。在此基础上，我们最终证明了自由交换fi-TD代数上的左可数双代数是连通的，并且是左可数λ-TD代数，从而是左可数fi代数。
A bstract . The theory of operated algebras has played a pivotal role in mathematics and physics. In this paper, we introduce a λ -TD algebra that appropriately includes both the Rota-Baxter algebra and the TD-algebra. The explicit construction of free commutative λ -TD algebra on a commutative algebra is obtained by generalized shu ffl e products, called λ -TD shu ffl e products. We then show that the free commutative λ -TD algebra possesses a left counital bialgera structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative λ -TD algebra is connected and filtered, and thus is a left counital Hopf algebra.