Noncommutative Differentials on Poisson-Lie groups and pre-Lie algebras

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 $\C[SU_2]\lrbicross U_\lambda(su_2^*)$.