Root systems, Nichols algebras and hypergeometric functions

We introduce Nichols algebras as certain combinatorially defined algebras associated to a braided vector space. In the two easiest examples, this produces symmetric algebras and exterior algebras.

Much more complicated are the examples that lead to the Borel part of quantum groups. In fact, every finite Nichols algebra comes with a generalized root system. I report on my recent proof, that certain hypergeometric functions have zeroes according to relations of an associated Nichols algebra. In current work, I prove a similar statement for solutions of a large class of differential equations involving multivalued complex functions. As an application, this proves that certain so-called screening operators in a conformal quantum field theory constitute an action of an associated Nichols algebra.