2024/01/24 by Foissy, Loïc, Patras, Frédéric
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.13317
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_∞ 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_∞ structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies.