Tridendriform structure on combinatorial Hopf algebras

TL;DR

This paper introduces a weighted extension of tridendriform bialgebras, establishing an equivalence with q-Gerstenhaber-Voronov algebras and demonstrating their natural occurrence in combinatorial Hopf algebras like surjective maps and parking functions.

Contribution

It extends tridendriform bialgebras with a weight q, defines q-Gerstenhaber-Voronov algebras, and links these structures to combinatorial Hopf algebras, providing new algebraic insights.

Findings

Equivalence between connected q-tridendriform bialgebras and q-Gerstenhaber-Voronov algebras.

Natural q-tridendriform structures on surjective maps and parking functions.

The M-permutations bialgebra as a quotient of ST(q).

Abstract

We extend the definition of tridendriform bialgebra by introducing a weight q. The subspace of primitive elements of a q-tridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called q-Gerstenhaber-Voronov algebras. We prove the equivalence between the categories of connected q-tridendriform bialgebras and of q-Gerstenhaber-Voronov algebras. The space spanned by surjective maps, as well as the space spanned by parking functions, have natural structures of q-tridendriform bialgebras, denoted ST(q) and PQSym(q)*, in such a way that ST(q) is a sub-tridendriform bialgebra of PQSym(q)*. Finally we show that the bialgebra of M-permutations defined by T. Lam and P. Pylyavskyy may be endowed with a natural structure of q-tridendriform algebra which is a quotient of ST(q).