Duality for Generalised Differentials on Quantum Groups and Hopf quivers

We study generalised differential structures $\Omega^1,d$ on an algebra $A$, where $A\tens A\to \Omega^1$ given by $a\tens b\to a d b$ need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs $(\Lambda^1,\omega)$ where $\Lambda^1$ is a right module and $\omega$ a right module map, and the Hopf algebra bicovariant case corresponds to morphisms $\omega:A^+\to \Lambda^1$ in the category of right crossed (or Drinfeld-Radford-Yetter) modules over $A$. When $A=U(g)$ the generalised left-covariant differential structures are classified by cocycles $\omega\in Z^1(g,\Lambda^1)$. We then introduce and study the dual notion of a codifferential structure $(\Omega^1,i)$ on a coalgebra and for Hopf algebras the self-dual notion of a strongly bicovariant differential graded algebra $(\Omega,d)$ augmented by a codifferential $i$ of degree -1. Here $\Omega$ is a graded super-Hopf algebra extending the Hopf algebra $\Omega^0=A$ and, where applicable, the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. We show how to construct such objects from first order data, with both a minimal construction using braided-antisymmetrizes and a maximal one using braided tensor algebras and with dual given via braided-shuffle algebras. The theory is applied to quantum groups with $\Omega^1(C_q(G))$ dually paired to $\Omega^1(U_q(g))$, and to finite groups in relation to (super) Hopf quivers.