# Seminario MALGA Padova Verona - Moduli Algebre Anelli: “Quiver Grassmannians of Dynkin type” + “Gradings on central division algebras”

## Venerdì 20 Maggio 2016, ore 15:00 - Aula 2AB45 - Giovanni Cerulli Irelli + Juan Cuadra

ARGOMENTI: Seminari

Venerdì 20 Maggio 2016 in Aula 2AB45, si terranno due seminari nell'ambito del Seminario Padova-Verona MALGA - Moduli Algebre Anelli.

ore 15:00 Giovanni Cerulli Irelli (Università di Roma La Sapienza)
“Quiver Grassmannians of Dynkin type“
Abstract
Given a Dynkin quiver Q and a finite dimensional complex Q-representation M, one can consider the variety of all subrepresentations of M. Such variety is stratified according to the dimension vectors of the subrepresentations and each strata is called a quiver Grassmannian. In this talk I will survey on recent results concerning the geometry of such quiver Grassmannians. In particular, I will provide another proof of the fact that quiver Grassmannians associated with exceptional representations have positive Poincaré polynomials and hence positive Euler characteristic (confirming a conjecture of Fomin and Zelevinsky). The talk is partially based on joint papers with M. Reineke and E. Feigin, and partially on the recent preprint arXiv: 1602.03039.

ore 16:30 Juan Cuadra (University of Almeria)
Question. Let $k$ be an algebraically closed field of characteristic zero. Let $D$ be a division algebra of degree $d$ over its center $Z(D)$. Assume that $k \subset Z(D)$. Which finite-dimensional Hopf algebras can act inner faithfully on $D$?
An action of a finite-dimensional Hopf algebra $H$ (over $k$) is called inner faithful if it does not factor through a smaller Hopf algebra $H/I$ for some nonzero Hopf ideal $I$ of $H$. It suffices to focus on such actions because any action factors through an inner faithful one. Not much is known about this question. The novelty with respect to previous investigations of the same kind is that the center is allowed to be an infinite extension of $k$ and that the action is not necessarily linear over
In this talk we will tackle this problem when $H$ is the function algebra $k^G$ for a finite group $G$. A faithful action of $k^G$ on $D$ is just a faithful grading by $G$. We will show that $G$ faithfully grades $D$ if and only if $G$ contains a normal abelian subgroup of index dividing $d$. This result is part of a joint work with Pavel Etingof (Massachusetts Institute of Technology) to appear in Int. Math. Res. Not. and available at arXiv.org.