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arXiv · 2208.05773

$λ$-TD algebras, generalized shuffle products and left counital Hopf algebras

Hengyi Luo·Shanghua Zheng
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Abstract

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 shuffle products, called $λ$-TD shuffle 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.

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Hengyi Luo, Shanghua Zheng. 2022-07-28. $λ$-TD algebras, generalized shuffle products and left counital Hopf algebras. https://arxiv.org/abs/2208.05773

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