The Bernstein homomorphism via Aguiar-Bergeron-Sottile universality

TL;DR

This paper constructs a canonical algebra homomorphism from a commutative connected graded Hopf algebra to its tensor product with quasisymmetric functions, extending known structures using the universality of QSym.

Contribution

It generalizes the Bernstein homomorphism for commutative graded Hopf algebras via the universal property of QSym, broadening its applicability.

Findings

Defines a canonical homomorphism H -> H (x) QSym.

Extends the internal comultiplication on QSym.

Utilizes the universal property of QSym for construction.

Abstract

If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H -> H (x) QSym is defined, where QSym denotes the Hopf algebra of quasisymmetric functions over k. This homomorphism generalizes the "internal comultiplication" on QSym, and extends what Hazewinkel (in Section 18.24 of his "Witt vectors") calls the Bernstein homomorphism. We construct this homomorphism with the help of the universal property of QSym as a combinatorial Hopf algebra (a well-known result by Aguiar, Bergeron and Sottile) and extension of scalars (the commutativity of H allows us to consider, for example, H (x) QSym as an H-Hopf algebra, and this change of viewpoint significantly extends the reach of the universal property).