# Complement to the Kernel of a Hopf Algebra Map

Given two Hopf algebras $A,B$ over a field $k$, and a Hopf algebra map $\pi:A \to B$, are there any general tricks for finding a complement to the kernel of $\pi$. That is, how can one find a subspace $C$ such that $C \oplus$ker$(\pi) = A$?

P.S. To confirm, by complement I mean I simply mean vector space complement.

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Let $A, B$ be Hopf algebras over a field $k$. What kind of complement are you looking for, i.e. should $C$ just be a $k$-vector space or should it be an $A$-module ? –  Ralph Jun 28 '11 at 15:49