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

Shahn Majid and Wen-Qing Tao

Vol. 284 (2016), No. 1, 213–256

DOI: 10.2140/pjm.2016.284.213

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 ${ℂ}_q\left[{SU}_2\right]$. At the noncommutative geometry level we show that the enveloping algebra $U\left(\mathfrak{m}\right)$ of a Lie algebra $\mathfrak{m}$, viewed as quantisation of ${\mathfrak{m}}^{\ast }$, admits a connected differential exterior algebra of classical dimension if and only if $\mathfrak{m}$ admits a pre-Lie algebra structure. We give an example where $\mathfrak{m}$ 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 $ℂ\left[{SU}_2\right]▸⊲U_{\lambda }\left({su}_2^{\ast }\right)$.

Keywords

noncommutative geometry, quantum group, left-covariant, differential calculus, bicovariant, deformation, Poisson–Lie group, pre-Lie algebra, (co)tangent bundle, bicrossproduct, bosonisation

Mathematical Subject Classification 2010

Primary: 17D25, 58B32, 81R50

Milestones

Received: 5 February 2015

Revised: 3 March 2016

Accepted: 9 March 2016

Published: 10 July 2016