Affine Nichols algebras of diagonal type and modular tensor categories

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Root systems and crystallographic Coxeter groups are central tools in the study of semisimple Lie algebras. In the structure theory of pointed Hopf algebras a similar role is expected to be played by Weyl groupoids and their root systems. Finite universal Weyl groupoids correspond to the crystallographic arrangements introduced by the applicant. These are arrangements of hyperplanes which satisfy a certain global axiom of integrality. In a series of papers, Heckenberger and the applicant achieved a complete classification of crystallographic arrangements up to isomorphisms. We propose to extend the results on Nichols algebras of diagonal type and Weyl groupoids to the affine case. We expect similar results as in the classical theory. In particular, there should also exist an exotic Fourier transform matrix connecting the combinatorics of crystallographic arrangements to certain modular tensor categories. Our second objective is to find for each finite Weyl groupoid W an associated Hopf algebra H having Was symmetry structure. We propose to approach this question by the study of the sheaf of (skew) differential operators on the toric variety of the corresponding crystallographic arrangement.