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

Abstract

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a [Formula: see text]-TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative [Formula: see text]-TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the [Formula: see text]-TD shuffle product. We then show that the free commutative [Formula: see text]-TD algebra possesses a left counital bialgebra 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 [Formula: see text]-TD algebra is connected and filtered, and thus is a left counital Hopf algebra.