  We show that the quantisation of a connected simply-connected Poisson-Lie
group admits a left-covariant noncommutative differential structure at lowest
deformation order if and only if the dual of its Lie algebra admits a pre-Lie
algebra structure. As an example, we find a pre-Lie algebra structure
underlying the standard 3D differential structure on $\C_q[SU_2]$. At the
noncommutative geometry level we show that the enveloping algebra $U(\cm)$ of a
Lie algebra $\cm$, viewed as quantisation of $\cm^*$, admits a connected
differential exterior algebra of classical dimension if and only if $\cm$
admits a pre-Lie algebra. We give an example where $\cm$ is solvable and we
extend the construction to the quantisation of tangent and cotangent spaces of
Poisson-Lie groups by using bicross-sum and bosonization of Lie bialgebras. As
an example, we obtain natural 6D left-covariant differential structures on the
bicrossproduct $\C[SU_2]\lrbicross U_\lambda(su_2^*)$.
