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[Gamma] for Gamma a finite abelian group, and with associated graded Hopf algebra of the form B(V) k[Gamma]YD. As a corollary to a B(V) # k[Gamma] where B(V) is the Nichols algebra of V = circle plus(i) V-gi(chii) is an element of (k[Gamma])(k[Gamma])yD. general theorem on duals of coradically graded Hopf algebras, we have that the dual of B(V) # k[Gamma] is B(W) # k[<(Gamma)over cap>] where W = circle plus(i) W-chii(gi) is an element of (k[<(Gamma)over cap>]YD)-Y-k[<(Gamma)over cap>]. This description of the dual is used to explicitly describe the Drinfel'd double of B(V) # k[F]. We also show that the dual of a nontrivial lifting A of B(V) # k[F] 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 B(V) # k[F]. We conclude with some examples where we determine all the irreducible finite-dimensional representations of a lifting of B(V) # k[Gamma] by computing the matrix coalgebras in the coradical of the dual. (C) 2003 Elsevier Science (USA). All rights reserved.