  Some duality properties for induced representations of enveloping algebras
involve the character $Trad_{\goth g}$. We extend them to deformation Hopf
algebras $A_{h}$ of a noetherian Hopf $k$-algebra $A_{0}$ satistying
$Ext^{i}_{A_{0}}(k, A_{0})=\{0\}$ except for $i=d$ where it is isomorphic to
$k$. These duality properties involve the character of $A_{h}$ defined by right
multiplication on the one dimensional free $k[[h]]$-module $Ext^{d}_{A_{h}}
(k[[h]], A_{h})$. In the case of quantized enveloping algebras, this character
lifts the character $Trad_{\goth g}$. We also prove Poincar{\'e} duality for
such deformation Hopf algebras in the case where $A_{0}$ is of finite
homological dimension. We explain the relation of our construction with quantum
duality.
