What is a cocentral homomorphism

Question

Suppose $A$ is a bialgebra, $H$ a Hopf algebra. A map $f:A \to H$ is a cocentral bialgebra homomorphism. What does it mean? What about central homomorphisms?

Answer 1

By definition, $f$ is cocentral if $$(id_A\otimes f)\circ \Delta_A = (id_A\otimes f)\circ \Delta_A^{op}$$ or with Sweedler notation, if for every $x\in A$: $$x_{(1)}\otimes f(x_{(2)}) = x_{(2)}\otimes f(x_{(1)}).$$

It is central if the image of $f$ lands in the Hopf center of $H$.