Duals of pointed Hopf algebras

In this paper, we study the duals of some finite-dimensional pointed Hopf algebras working over an algebraically closed field k of characteristic 0. In particular, we study pointed Hopf algebras with coradical k[Γ] for Γ a finite abelian group, and with associated graded Hopf algebra of the form where is the Nichols algebra of . As a corollary to a general theorem on duals of coradically graded Hopf algebras, we have that the dual of is where . This description of the dual is used to explicitly describe the Drinfelʹd double of . We also show that the dual of a nontrivial lifting A of which is not itself a Radford biproduct, is never pointed. For V a quantum linear space of dimension 1 or 2, we describe the duals of some liftings of . We conclude with some examples where we determine all the irreducible finite-dimensional representations of a lifting of by computing the matrix coalgebras in the coradical of the dual.