  We give an analogue of the classical exponential map on Lie groups for Hopf
$*$-algebras with differential calculus. The major difference with the
classical case is the interpretation of the value of the exponential map,
classically an element of the Lie group. We give interpretations as states on
the Hopf algebra, elements of a Hilbert $C^{*} $-bimodule of $\frac{1}{2}$
densities and elements of the dual Hopf algebra. We give examples for complex
valued functions on the groups $S_{3}$ and $\mathbb{Z}$, Woronowicz's matrix
quantum group $\mathbb{C}_{q}[SU_2] $ and the Sweedler-Taft algebra.
