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-ie algebra structure. As an example, we find a pre-Lie algebra structure underlying the standard 3-dimensional differential structure on C-q [SU2]. At the noncommutative geometry level we show that the enveloping algebra U (m) of a Lie algebra m, viewed as quantisation of m*, admits a connected differential exterior algebra of classical dimension if and only if m admits a pre-Lie algebra structure. We give an example where m is solvable and we extend the construction to tangent and cotangent spaces of Poisson-Lie groups by using bicross-sum and bosonisation of Lie bialgebras. As an example, we obtain a 6-dimensional left-covariant differential structure on the bicrossproduct quantum group C[SU2] proportional to U-lambda (su(2)*).