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

Abstract

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 3 -dimensional differential structure on C q [ SU 2 ] . 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 [ SU 2 ] ▶ ⊲ U λ ( su ∗ 2 ) .