A combinatorial Hopf algebra for the boson normal ordering problem

Bousbaa, Imad Eddine; Chouria, Ali; Luque, Jean-Gabriel

Annales de l’Institut Henri Poincaré D, Tome 5 (2018) no. 1, pp. 61-102.

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.

Classification : 05-XX, 16-XX, 81-XX
Mots-clés : Normal boson ordering, Fock space, generalized Stirling numbers, combinatorial Hopf algebras