A Hopf algebra on posets extending the Connes–Kreimer Hopf algebra

Abstract. We construct a graded connected Hopf algebra H(X,K) associatedwith any finite poset (X,⪯) over a commutative ring K. The underlying algebra isthe commutative polynomial ring in variables indexed by the nonempty subsetsof X, equipped with a coproduct defined via ideal decompositions: ∆(yσ) =I∈I(σ) yσ∖I ⊗ yI, where I(σ) is the set of ideals of (σ,⪯). We prove thatthe group of characters of H(X,K) is naturally isomorphic to SL(X,K)op, theopposite of the group of units with augmentation one in the algebra of filtersAF(X,K); in the rooted forest case this recovers the Butcher group. We definea connected quotient Hc(X,K) and show that when X is a rooted forest withthe tree ordering, the coproduct on Hc(X,K) coincides with the Connes–Kreimercoproduct. We compute the antipode recursively, determine the space of primitiveelements, and show that H(X,K) is commutative but not cocommutative ingeneral. Finally, we show that the assignment (X,⪯) → H(X,K) is functorialwith respect to order embeddings of posets, with the character group functorcontravariantly sending each poset to SL(X,K)op