Left counital Hopf algebras on bi-decorated planar rooted forests and Rota-Baxter systems

Abstract

Involving an extended 1-cocycle condition, we first define a coproduct on the space of bi-decorated planar rooted forests to equip it with a left counital bialgebraic structure. We introduce the concept of left counital $(\Omega, \alpha)$-cocyle bialgebras and show that the space of bi-decorated planar rooted forests is the free object in the category of left counital $(\Omega, \alpha)$-cocyle bialgebras. We then prove a generalized fact that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is only right-sided. Having this fact in hand, a left counital Hopf algebraic structure on bi-decorated rooted forests is also established. Finally, we construct a Rota-Baxter system on the left counital Hopf algebra.