Noncommutative connections on bimodules and Drinfeld twist deformation

Paolo Aschieri, Alexander Schenkel

Research output: Contribution to journalArticlepeer-review

32 Citations (Scopus)


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 Hequivariant 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.

Original languageEnglish
Pages (from-to)513-612
Number of pages100
JournalAdvances in Theoretical and Mathematical Physics
Issue number3
Publication statusPublished - 2014

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Mathematics(all)


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