A geometric approach to Nichols algebras and their approximations

Nichols algebras, also known as small shuffle algebras, are a family of graded bialgebras including the symmetric algebras, the exterior algebras, the positive parts of quantized enveloping algebras, and, conjecturally, Fomin-Kirillov algebras. As the case of Fomin-Kirillov algebra shows, it can be very
difficult to determine the maximum degree of a minimal generating set of relations of a Nichols algebra. 

Building upon Kapranov and Schechtman’s equivalence between the category of perverse sheaves on Sym(C) and the category of graded connected bialgebras,  we describe the geometric counterpart of the maximum degree of a generating set of relations of a graded connected bialgebra, and we show how this specialises to the case o Nichols algebras.

The talk is based on joint work with Francesco Esposito and Lleonard Rubio y Degrassi.