# Annales de l’Institut Henri Poincaré D

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**Volume 5, Issue 1, 2018, pp. 61–102**

**DOI: 10.4171/AIHPD/48**

Published online: 2018-02-01

A combinatorial Hopf algebra for the boson normal ordering problem

Imad Eddine Bousbaa^{[1]}, Ali Chouria

^{[2]}and Jean-Gabriel Luque

^{[3]}(1) USTHB, Alger, Algeria

(2) Université de Rouen, Saint-Étienne-du-Rouvray, France

(3) Université de Rouen, Saint-Étienne-du-Rouvray, France

In the aim of understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \emph{et al.} defined combinatorial objects allowing to interpret the number of $S_{\bf{r,s}}(k)$ appearing in the identity $(a^\dagger)^{r_n}a^{s_n}\cdots(a^\dagger)^{r_1}a^{s_1}=(a^\dagger)^\alpha\displaystyle\sum S_{\bf{r,s}}(k)(a^\dagger)^k a^k$, where $\alpha$ is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which projects to the enveloping algebra of the Heisenberg Lie algebra. Here, we propose a new variant this construction which admits a realization with variables. This means that we construct our algebra from a free algebra $\mathbb{C}\langle A \rangle$ using quotient and shifted product. The combinatorial objects (B-diagrams) are slightly different from those proposed by Blasiak \emph{et al.}, but give also a combinatorial interpretation of the generalized Stirling numbers together with a combinatorial Hopf algebra related to Heisenberg Lie algebra. the main difference comes the fact that the B-diagrams have the same number of inputs and outputs. After studying the combinatorics and the enumeration of B-diagrams, we propose two constructions of algebras called. The Fusion algebra $\mathcal{F}$ defined using formal variables and another algebra $\mathcal{B}$ constructed directly from the B-diagrams. We show the connection between these two algebras and that $\mathcal{B}$ can be endowed with Hopf structure. We recognise two already known combinatorial Hopf subalgebras of $\mathcal{B}$: WSym the algebra of word symmetric functions indexed by set partitions and BWSym the algebra of biword symmetric functions indexed by set partitions into lists.

*Keywords: *Normal boson ordering, Fock space, generalized Stirling numbers, combinatorial Hopf algebras

Bousbaa Imad Eddine, Chouria Ali, Luque Jean-Gabriel: A combinatorial Hopf algebra for the boson normal ordering problem. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 5 (2018), 61-102. doi: 10.4171/AIHPD/48