Rigid DAHA modules: the case of A1

We will begin with the definitions; only type A1 will be considered. Then we will switch to roots of unity and construct 2 major series of perfect representations, where the simplest one (upon t=q and the symmetrization) is the Verlinde algebra.
The 3rd series is non-semisimple, related to logarithmic conformal field theory. All of them are rigid, which guarantees the action of projective PSL(2,Z) and the absolute Galois group (over Q). And all of them are challenges for theory of tensor categories.
If time permits, we will use this classification to outline connections with discrete Mostow-Livne subgroups in PU(2,1).