Theta maps for combinatorial Hopf algebras

TL;DR

This paper introduces generalized Theta maps for combinatorial Hopf algebras, providing a framework to identify odd Hopf subalgebras and describing specific Theta maps for various algebraic structures.

Contribution

It generalizes the concept of Theta maps to arbitrary combinatorial Hopf algebras and offers criteria and combinatorial descriptions for these maps.

Findings

Constructed generalized Theta maps for various Hopf algebras.

Provided a strategy to find odd Hopf subalgebras.

Described Theta maps for specific algebraic structures.

Abstract

There is a very natural and well-behaved Hopf algebra morphism from quasisymmetric functions to peak algebra, which we call it Theta map. This paper focuses on generalizing the peak algebra by constructing generalized Theta maps for an arbitrary combinatorial Hopf algebra. The image of Theta maps lies in the odd Hopf subalgebras, so we present a strategy to find odd Hopf subalgebra of any combinatorial Hopf algebra. We also give a combinatorial description of a family of Theta maps for Malvenuto-Reutenauer Hopf algebra of permutations $\operatorname{\mathsf{\mathfrak{S}Sym}}$ whose images are generalizations of the peak algebra. We also indicate a criterion to check whether a map is a Theta map. Moreover, precise descriptions of the Theta maps for the following Hopf algebras will be presented, Hopf subalgebras of quasisymmetric functions, commutative and co-commutative Hopf algebras, and theta maps for a Hopf algebra $\mathcal{V}$ on permutations.