The finite dual $H^{ \circ}$ of an affine commutative-by-finite Hopf algebra $H$ is studied. Such a Hopf algebra $H$ is an extension of an affine commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main theorem gives natural conditions under which $H^{ \circ}$ decomposes as a crossed or smash product of $F^{ \ast}$ by the finite dual of $A$. This decomposition is then further analysed using the Cartier- Gabriel-Kostant theorem to obtain component Hopf subalgebras of $H^{ \circ}$ mapping onto the classical components of $A^{ \circ}$. The detailed consequences for a number of families of examples are then studied.
Categories: math.QA math.RA