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 B(V )#k[Γ] where B(V ) is the Nichols algebra of V = ⊕iV χi gi ∈ k[Γ] k[Γ]YD. As a corollary to a general theorem on duals of coradically graded Hopf algebras, we have that the dual of B(V )#k[Γ] is B(W )#k[Γ̂] where W = ⊕iW gi χi ∈ k[Γ̂] k[Γ̂] YD. This description of the dual is used to explicitly describe the Drinfel’d double of B(V )#k[Γ]. We also show that the dual of a nontrivial lifting A of B(V )#k[Γ] 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[Γ]. We conclude with some examples where we determine all the irreducible finite-dimensional representations of a lifting of B(V )#k[Γ] by computing the matrix coalgebras in the coradical of the dual.