Noncommutative connections on bimodules and Drinfeld twist deformation

Aschieri, Paolo; Schenkel, Alexander

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.

• Journal Article Type: Article
• Acceptance Date: Oct 1, 2014
• Online Publication Date: Oct 29, 2014
• Publication Date: Oct 29, 2014
• Deposit Date: Aug 22, 2019
• Publicly Available Date: Sep 4, 2019
• Journal: Advances in Theoretical and Mathematical Physics
• Print ISSN: 1095-0761
• Electronic ISSN: 1095-0753
• Publisher: International Press
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 3
• Pages: 513-612
• Keywords: Quantum Algebra; High Energy Physics - Theory; Mathematical Physics
• Public URL: https://nottingham-repository.worktribe.com/output/2460539
• Publisher URL: https://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0018/0003/a001/index.php