Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables

TL;DR

This paper develops a coloured generalization of non-commutative symmetric functions and their dual quasi-symmetric functions, providing explicit formulas and structural insights in the context of partially commutative variables and rooted trees.

Contribution

It introduces a new Hopf algebra structure for coloured non-commutative symmetric functions and their duals, expanding the algebraic framework to partially commutative variables and rooted trees.

Findings

Defined the algebra $ ext{NSym}_A$ as a subalgebra of rooted ordered coloured trees.

Constructed the dual algebra $ ext{QSym}_A$ of coloured quasi-symmetric functions with power series realization.

Derived formulas for multiplication, comultiplication, and antipode in various bases.

Abstract

We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases -- the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.