Semi-classical differential structures

TL;DR

This paper develops a semiclassical approach to differential calculus in noncommutative geometry, linking symplectic connections, curvature, and quantisation, with applications to quantum groups and the noncommutative torus.

Contribution

It introduces a semiclassical framework for differential calculus on noncommutative spaces, characterizes symplectic connections, and explores moduli spaces of infinitesimal data for quantum groups.

Findings

Symplectic case links infinitesimal data to symplectic connections.

Moduli space of bicovariant infinitesimal data is characterized for simple Lie algebras.

Quantisation relates the canonical preconnection to Drinfeld twists.

Abstract

We semiclassicalise the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups. We show in the symplectic case that the infinitesimal data for a differential calculus is a symplectic connection, and interpret its curvature as lowest order nonassociativity of the exterior algebra. Semiclassicalisation of the noncommutative torus provides an example with zero curvature. In the Poisson-Lie group case we study left-covariant infinitesimal data in terms of partially defined preconnections. We show that the moduli space of bicovariant infinitesimal data for quasitriangular Poisson-Lie groups has a canonical reference point which is flat in the triangular case. Using a theorem of Kostant, we completely determine the moduli space when the Lie algebra is simple: the canonical preconnection is the unique point for other than sl_n, n>2, when the moduli space is 1-dimensional. We relate the canonical preconnection to Drinfeld twists and thereby quantise it to a super coquasi-Hopf exterior algebra. We also discuss links with Fedosov quantisation.