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

The theory of operated algebras has played a pivotal role in mathematics and physics. In this paper, we introduce a $\lambda$-TD algebra that appropriately includes both the Rota-Baxter algebra and the TD-algebra. The explicit construction of free commutative $\lambda$-TD algebra on a commutative algebra is obtained by generalized shuffle products, called $\lambda$-TD shuffle products. We then show that the free commutative $\lambda$-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 $\lambda$-TD algebra is connected and filtered, and thus is a left counital Hopf algebra.