Hopf algebras on decorated noncrossing arc diagrams

TL;DR

This paper introduces a general method to construct Hopf algebra structures on decorated noncrossing arc diagrams, linking combinatorial models with algebraic frameworks and recovering known Hopf algebras.

Contribution

It provides a unified approach to endow decorated noncrossing arc diagrams with Hopf algebra structures, encompassing previously known cases.

Findings

Established a general method for Hopf algebra structures

Recovered known Hopf algebras as special cases

Bridged combinatorial models with algebraic structures

Abstract

Noncrossing arc diagrams are combinatorial models for the equivalence classes of the lattice congruences of the weak order on permutations. In this paper, we provide a general method to endow these objects with Hopf algebra structures. Specific instances of this method produce relevant Hopf algebras that appeared earlier in the literature.