Duality for generalised differentials on quantum groups

We study generalised differential structures (Ω1,d)(Ω1,d) on an algebra A  , where A⊗A→Ω1A⊗A→Ω1 given by a⊗b→adba⊗b→adb need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs (Λ1,ω)(Λ1,ω) where Λ1Λ1 is a right module and ω   a right module map, and the Hopf algebra bicovariant case corresponds to morphisms ω:A+→Λ1ω:A+→Λ1 in the category of right crossed (or Drinfeld–Radford–Yetter) modules over A  . When A=U(g)A=U(g) the generalised left covariant differential structures are classified by cocycles ω∈Z1(g,Λ1)ω∈Z1(g,Λ1). We then introduce and study the dual notion of a codifferential structure (Ω1,i)(Ω1,i) on a coalgebra and for Hopf algebras the self-dual notion of a strongly bicovariant differential graded algebra (Ω,d)(Ω,d) augmented by a codifferential i   of degree −1. Here Ω is a graded super-Hopf algebra extending the Hopf algebra Ω0=AΩ0=A and, where applicable, the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. Accordingly, group 1-cocycles correspond precisely to codifferential structures on algebraic groups and function algebras. Among general constructions, we show that first order data (Λ1,ω)(Λ1,ω) on a Hopf algebra A   extends canonically to a strongly bicovariant differential graded algebra via the braided super-shuffle algebra. The theory is also applied to quantum groups with Ω1(Cq(G))Ω1(Cq(G)) dually paired to Ω1(Uq(g))Ω1(Uq(g)).