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

Ali Chouria, Jean-Gabriel Luque

TL;DR

This work introduces the Hopf algebra of B-diagrams as a diagrammatic framework encoding bosonic normal ordering and projecting onto the Heisenberg algebra, while clarifying its deep connections to the noncommutative symmetric functions and to colored set partitions. It develops the dual Hopf algebra structure, establishes a multiplicative basis, and shows how WSym, BWSym, and related colored partition algebras arise as subalgebras or quotients, illustrating the unifying role of B-diagrams in combinatorial Hopf algebras. Through a sequence of explicit constructions and examples, the paper demonstrates a rich duality between diagrammatic objects and classical combinatorial Hopf algebras, with potential universality across diagrammatic models in physics and algebra. The results position the B-diagram Hopf algebra as a natural, versatile tool for representing normal ordering problems and for bridging diverse combinatorial Hopf algebra families, inviting further study of its representation theory and universal properties.

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.

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

TL;DR

This work introduces the Hopf algebra of B-diagrams as a diagrammatic framework encoding bosonic normal ordering and projecting onto the Heisenberg algebra, while clarifying its deep connections to the noncommutative symmetric functions and to colored set partitions. It develops the dual Hopf algebra structure, establishes a multiplicative basis, and shows how WSym, BWSym, and related colored partition algebras arise as subalgebras or quotients, illustrating the unifying role of B-diagrams in combinatorial Hopf algebras. Through a sequence of explicit constructions and examples, the paper demonstrates a rich duality between diagrammatic objects and classical combinatorial Hopf algebras, with potential universality across diagrammatic models in physics and algebra. The results position the B-diagram Hopf algebra as a natural, versatile tool for representing normal ordering problems and for bridging diverse combinatorial Hopf algebra families, inviting further study of its representation theory and universal properties.

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.
Paper Structure (30 sections, 11 theorems, 162 equations, 11 figures)

This paper contains 30 sections, 11 theorems, 162 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background and preliminary results on Hopf algebras
  3. Bialgebras and Hopf algebras
  4. Compatibility of the unit and counit.
  5. Duality of Hopf Algebras.
  6. The Specific Type of Hopf Algebra Considered
  7. Examples of combinatorial Hopf algebras
  8. Non commutative polynomials
  9. Word symmetric Hopf algebra
  10. Bi-words symmetric functions
  11. Cocommutative Hopf algebras of noncommutative polynomials
  12. The Hopf algebra of B-diagrams
  13. What B-diagrams are
  14. The Hopf algebra of B-diagrams
  15. Algebraic structure
  16. ...and 15 more sections

Key Result

Lemma 1

The product $u_1\Cup u_2$ of two words $u_1,u_2\in\Sigma^*$ is inductively defined by

Figures (11)

  • Figure 1: Commutative diagram
  • Figure 2: Commutative diagram characterizing the antipode of a Hopf algebra.
  • Figure 3: The $B$-diagram $(4,[3,2,2,1],7486\sqcup\sqcup\sqcup\sqcup,\{5,6\}, \{1,2,3,5\})$ .
  • Figure 4: An example of composition when $k=0$
  • Figure 5: The $B$-diagram $G[1,3]$ of $G= (4,[3,2,2,1],7486\sqcup\sqcup\sqcup\sqcup,\{5,6\}, \{1,2,3,5\})$
  • ...and 6 more figures

Theorems & Definitions (45)

  • Lemma 1
  • Example 2
  • Example 3
  • Lemma 4
  • Example 5
  • Proposition 6
  • Definition 7
  • Example 8
  • Definition 9
  • Example 10
  • ...and 35 more