On the Connes-Kreimer construction of Hopf algebras (Q2752879)

The complex algebra \(H\) of rooted trees is the polynomial algebra in countably many indeterminates, one for each finite rooted tree. \(H\) has a comultiplication making it into a Hopf algebra, and a linear endomorphism \(\lambda\) which is a universal cocycle for an appropriate Hochschild cohomology of Hopf algebras. The author notes that this structure can be deduced from the fact that \(H\) is the initial object in the category of commutative unitary algebras equipped with a linear endomorphism. From this one sees that \(H\) has many Hopf algebra structures all with \(\lambda\) as universal cocycle. Examples of such comultiplications are given for any two complex numbers \(q_1\) and \(q_2\). For \(q_1=1\) and \(q_2=0\), one recovers the original Hopf algebra structure on \(H\). All this is done as an example of a more general universal construction of families of Hopf \(\mathbb{P}\)-algebras for any Hopf operad \(\mathbb{P}\). More precisely, if \(\mathbb{P}\) is a Hopf operad on a symmetric monoidal additive category, then the initial object in the category of \(\mathbb{P}\)-algebras with a linear endomorphism has a family of Hopf \(\mathbb{P}\)-algebra structures. Then the algebra of rooted trees is the extreme instance of this construction, where \(\mathbb{P}\) is the unit object in each degree.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].