Classification of differentials on quantum doubles and finite noncommutative geometry

We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the `quantum groups approach’. We apply the author’s previous classification theorem, implying that calculi in the factorisable case correspond to blocks in the dual, to classify differential calculi on the quantum codouble $D^{\ast }(G)=kG\text{lcocross}k(G)$ of a finite group $G$. We give $D^{\ast }(S_3)$ as an example including its exterior algebra and lower cohomology. We also study the calculus on $D^{\ast }(\text{CA})$ induced from one on a general Hopf algebra $\text{CA}$ in general and specialise to $D^*(G)=U(\cg)\lcocross k[G]$asanoncommutativeisometrygroupofanenvelopingalgebra$U(\cg)$ as a noncommutative space.