Some duality properties for
induced representations of enveloping algebras involve the character Tradg. We
extend them to deformation Hopf algebras Ah of a noetherian Hopf k-algebra A0
satisfying ExtA0i(k,A0) = {0} except for i = d where it is isomorphic to k. These
duality properties involve the character of Ah defined by right multiplication on the
one-dimensional free k[[h]]-module ExtAhd(k[[h]],Ah). In the case of quantized
enveloping algebras, this character lifts the character Tradg. We also prove Poincaré
duality for such deformation Hopf algebras in the case where k[[h]] is an Ah-module of
finite projective dimension. We explain the relation of our construction with quantum
duality.
