Differential Calculus in Braided Abelian Categories

AbstractBraided non-commutative differential geometry is studied. In particular we investigate the theory of (bi-covariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopfbimodules in braided categories are used to construct higher order bicovariant differential calculi over braidedHopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf alge-bras with universal bialgebra properties. The article especially extends Woronowicz’s results on (bicovariant)differential calculi to the braided non-commutative case. Keywords: Braided category, Hopf algebra, Hopf bimodule, Differential calculus 1991 Mathematics Subject Classification: 16W30, 18D10, 18E10, 17B37 Introduction Differential geometry and group theory interact very fruitfully within the theory of Lie groups. Tangent Liealgebras, invariant differential forms, infinitesimal representations, principal bundles, gauge theory, etc. emergedout of this interplay. It was discovered by Woronowicz that many of these geometrically related structures can begeneralized to non-commutative geometry [Wor]. He introduced and studied the differential calculus on compactquantum groups or more generally on Hopf algebras over a field k with char(k) = 0. His theory is built on the basecategory of k-modules with the usual tensor product and the involutive tensor transposition τ: a⊗ b→ b⊗ a. Inwhat follows we refer to the conditions in [Wor] as the classical conditions in contrast to our investigations in thebraided case. There have been a lot of publications along the classical lines of [Wor]. However, it would be beyondthe scope of this introduction to give an appropriate appreciation to all of them. Nevertheless we would like tomention three articles [Brz, Mal, Man] besides [Wor] which particularly influenced our work from the differentialgeometrical point of view in a considerable way. The differential graded algebra approach to quantum groupsand the quantum de Rham complexes were studied in [Mal, Man]. In Manin’s work [Man, Proposition 2.6.1] verygeneralconditions are found under which an operatorring (of differentials) overan algebrahas a bialgebrastructure.The non-commutative differential calculus discovered by Woronowicz is indeed of this type. The differential Hopfalgebra structure of the higher order differential calculi of [Wor] has been unfolded explicitely in [Brz]. Essentiallyall of these articles proceed on the assumption that the classical symmetric conditions are at the bottom of thetheory of (bicovariant) differential calculi.Braided or quasisymmetric monoidal categories had been initially investigated by Joyal and Street in [JS].Footing on their work, Majid generalized the notion of (tensor) algebras, bi- and Hopf algebras to categories with a