Hopf Algebras of B-Diagrams and Boson Normal Ordering: Exploring the Dual Structures

TL;DR

This paper investigates the Hopf algebra structure of B-diagrams related to bosonic normal ordering, revealing dualities and connections with combinatorial Hopf algebras to deepen understanding of noncommutative symmetric polynomials.

Contribution

It introduces and analyzes the dual Hopf algebra of B-diagrams, generalizing properties of noncommutative symmetric polynomials and linking to colored set partition Hopf algebras.

Findings

Dual Hopf algebra of B-diagrams is constructed and studied.

Connections established between B-diagrams and combinatorial Hopf algebras.

Generalization of properties of noncommutative symmetric polynomials.

Abstract

We consider the Hopf algebra of B-diagrams as an algebra projecting onto the Heisenberg algebra and designed to encode the combinatorics of the bosonic normal-ordering problem. In order to understand and generalize the properties of the algebra of noncommutative symmetric polynomials viewed as a Hopf subalgebra of the Hopf algebra linearly spanned by B-diagrams, we describe and study its dual Hopf algebra. This construction also allows us to establish connections with combinatorial Hopf algebras based on colored set partitions.