Noncommutative connections on bimodules and Drinfeld twist deformation

TL;DR

This paper develops a framework for noncommutative connections on bimodules influenced by Hopf algebra actions, extending deformation theory via Drinfeld twists to non-equivariant cases and tensor structures.

Contribution

It introduces a method to induce connections on tensor products of bimodules under Hopf algebra actions, extending deformation quantization to non-equivariant morphisms and connections.

Findings

Connections on bimodules can be induced from individual bimodules.

The framework applies to deformation quantization via Drinfeld twists.

The theory extends to non-equivariant morphisms and tensor algebras.

Abstract

Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over A of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily H-equivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra H is the universal enveloping algebra of vector fields (or a finitely generated Hopf subalgebra). We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily H-equivariant. The theory canonically lifts to the tensor product structure.