Commutative B\_infty -algebras are shuffle algebras

Abstract: We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative $B_\infty$ structure that governs the product and the eulerian idempotent, as well as the canonical projection on the space $V$. This generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle algebras, which correspond to the case when the $B_\infty$ structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies.