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

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.