The Bernstein homomorphism via Aguiar-Bergeron-Sottile universality
نویسنده
چکیده
If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H → H⊗QSymk is defined, where QSymk denotes the Hopf algebra of quasisymmetric functions. This homomorphism generalizes the “internal comultiplication” on QSymk, and extends what Hazewinkel (in §18.24 of his “Witt vectors”) calls the Bernstein homomorphism. We construct this homomorphism with the help of the universal property of QSymk as a combinatorial Hopf algebra (a well-known result by Aguiar, Bergeron and Sottile) and extension of scalars (the commutativity of H allows us to consider, for example, H ⊗QSymk as an H-Hopf algebra, and this change of viewpoint significantly extends the reach of the universal property).
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