Duality for Generalised Differentials on Quantum Groups and Hopf quivers

We introduce for Hopf algebras a self-dual notion of strongly bicovariant differential graded algebra (Ω,d) augmented by a codifferential i of degree -1. There is an associated derivation L=d i+i d. Here bicovariance is expressed as a graded super-Hopf algebra structure on Ω extending the Hopf algebra Ω^0 and where applicable the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. Moreover, the theory is most natural in a generalised setting where not every element of Ω^1 need be a sum of elements of the form a.db and we study and classify such generalised differentials. For finite sets they correspond to quivers with embedded digraphs, while bicovariant ones on a Hopf algebra A correspond to pairs (Λ^1,ω) where Λ^1 an object in the braided category of crossed (or Drinfeld-Radford-Yetter) modules over A and ω:A^+\to Λ^1 is a morphism, where the augmentation ideal A^+ is an object in this category by right multiplication and the adjoint coaction. A bicovariant codifferential is likewise given by a morphism i:Λ^1\to A^+ where A^+ has a complementary crossed module structure given by the adjoint action and the coproduct. We show how to construct augmented strongly bicovariant calculi (Ω,d,i) from first order data. The theory is applied at first order to quantum groups where the standard Ω^1(C_q(G)) are dually paired to certain Ω^1(U_q(g)) and arise naturally as generalised calculi not requiring factorisability of the quantum group. The theory is also applied to obtain the noncommutative extension by the Laplacian of the classical Ω(G) as dual to a certain noncommutative calculus Ω(U(g)). We relate strongly bicovariant calculi in the finite group case to Hopf quivers.