Duals of pointed Hopf algebras

Abstract

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=⊕ i V χ i g i ∈ 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[ Γ ] where W=⊕ i W g i χ i ∈ k[ Γ ] k[ Γ ] 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.