### Italian Representation Theory Seminar

The talk is based on work in progress with Francesco Esposito and Lleonard Rubio y Degrassi. Recently Kapranov and Schechtman have settled an equivalence between the category of graded connected co-connected bialgebras in a braided monoidal category $\mathcal V$ and the category of factorizable systems of perverse sheaves on all symmetric products ${\mathsf{Sym}}^n({\mathbb C})$ with values in $\mathcal V$. The Nichols (shuffle) algebra associated with an object $V$ corresponds to the system of intersection cohomology extensions of a precise local system on the open strata. Motivated by the study of Fomin-Kirillov algebras and their relation with Nichols algebras, we describe the factorizable perverse sheaves counterpart of some algebraic constructions, including the n-th approximation of a graded bialgebra, and we translate into geometric statements when a Nichols algebra is quadratic.