# A combinatorial Hopf algebra for the boson normal ordering problem

### Imad Eddine Bousbaa

USTHB, Alger, Algeria### Ali Chouria

Université de Rouen, Saint-Étienne-du-Rouvray, France### Jean-Gabriel Luque

Université de Rouen, Saint-Étienne-du-Rouvray, France

## Abstract

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 $S_{r,s}(k)$ appearing in the identity $(a_{†})_{r_{n}}a_{s_{n}}⋯(a_{†})_{r_{1}}a_{s_{1}}=(a_{†})_{α}∑S_{r,s}(k)(a_{†})_{k}a_{k}$, where $α$ 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 $C⟨A⟩$ 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 $F$ defined using formal variables and another algebra $B$ constructed directly from the B-diagrams. We show the connection between these two algebras and that $B$ can be endowed with Hopf structure. We recognise two already known combinatorial Hopf subalgebras of $B$: 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.

## Cite this article

Imad Eddine Bousbaa, Ali Chouria, Jean-Gabriel Luque, A combinatorial Hopf algebra for the boson normal ordering problem. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), no. 1, pp. 61–102

DOI 10.4171/AIHPD/48