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

TL;DR

This paper establishes a deep connection between pre-Lie algebra structures and the existence of covariant noncommutative differential structures on quantised Poisson-Lie groups, providing explicit examples and constructions.

Contribution

It demonstrates that pre-Lie algebra structures are necessary and sufficient for covariant differential calculus on quantised Poisson-Lie groups and their associated enveloping algebras.

Findings

Pre-Lie structures characterize covariant differential structures on quantised groups.

Explicit example on $ ext{C}_q[SU_2]$ with pre-Lie algebra underlying its differential structure.

Construction of differential structures on bicrossproduct quantum groups.

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