The Hopf algebras of type B quasisymmetric functions and peak functions

TL;DR

This paper establishes that type B quasisymmetric functions and peak functions can be structured as Hopf algebras with suitable coproducts, expanding the algebraic framework for these functions.

Contribution

It introduces a specific coproduct making type B quasisymmetric functions and peak functions into Hopf algebras and subalgebras, respectively.

Findings

Type B quasisymmetric functions form a Hopf algebra with the right coproduct.

Type B peak functions constitute a Hopf subalgebra.

Provides algebraic structure for type B functions.

Abstract

We show that with the appropriate choice of coproduct, the type B quasisymmetric functions form a Hopf algebra, and the recently introduced type B peak functions form a Hopf subalgebra.