Approximations of a Nichols algebra from a geometric point of view

by Giovanna Carnovale (Università degli Studi di Padova, Italy)

Thursday 9 Dec 2021, 15:00 → 16:00 Europe/Rome

Sala Seminari (Dipartimento di Matematica)

​The talk is based on an ongoing joint project with Francesco  Esposito and Lleonard Rubio y Degrassi.

Nichols (shuffle) algebras are a family of graded Hopf algebras (in a braided monoidal category) which includes symmetric algebras, exterior algebras, and the positive part of quantized enveloping algebras. They are crucial in the classification of (pointed) Hopf algebras. However,  it is very difficult to describe their relations or to estimate their dimensions in general and new tools are very welcome. We will exploit an equivalence due to  Kapranov and Schechtman between a category of graded bialgebras in a braided monoidal category $V$ and the category of factorizable systems of perverse sheaves on all symmetric products ${\mathsf{Sym}}^n(\mathbb{C})$ with values in $V$. I will describe the factorizable perverse sheaves counterpart of some algebraic constructions, including the $n$-th approximation of a graded bialgebra. Since the image of Nichols algebras through Kapranov and Schechtman equivalence is very precise, we can translate into geometric statements when a Nichols algebra is finitely presented, or coincides with any of its approximations.