For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra.In this article we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.

It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.

In this article, we explicitly construct new finite-dimensional, indecomposable Nichols algebras with Dynkin diagrams of type An, Cn, Dn, E6,7,8, F4 over any group G with commutator subgroup isomorphic to Z2. The construction is generic in the sense that the type just depends on the rank and center of G, and thus positively answers for all groups of this class a question raised by Susan Montgomery in 1995 [Mon95][AS02]. Our construction uses the new notion of a covering Nichols algebra as a special case of a covering Hopf algebra [Len12] and produces nonfaithful Nichols algebras. We give faithful examples of Doi twists for type A3, C3, D4, C4, F4 over several nonabelian groups of order 16 and 32. These are hence the first known examples of nondiagonal, finitedimensional, indecomposable Nichols algebras of rank > 2 over nonabelian groups.

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We prove a long-standing conjecture by B. Feigin et al. that certain screening operators on a conformal field theory obey the algebra relations of the Borel part of a quantum group (and more generally a diagonal Nichols algebra). Up to now this has been proven only for the quantum group uq(sl2).The proof is based on a novel, intimate relation between Hopf algebras, Vertex algebras and a class of analytic functions in several variables, which are generalizations of Selberg integrals. These special functions have zeroes wherever the associated diagonal Nichols algebra has a relation, because we can prove analytically a quantum symmetrizer formula for them. Morevover, we can use the poles of these functions to construct a crucial Weyl group action.Our result produces an infinite-dimensional graded representation of any quantum group or Nichols algebra. We discuss applications of this representation to Kazhdan-Lusztig theory.