# Geometric perspective on Nichols algebras

## TL;DR

This paper presents a geometric framework for understanding the generation of finite-dimensional pointed Hopf algebras, showing that Nichols algebras correspond to closed orbits under a reductive group action, and explores their rigidity and deformations.

## Contribution

It introduces a geometric perspective on Nichols algebras within braided categories, linking algebraic generation to orbit rigidity and deformation theory.

## Key findings

- Nichols algebras correspond to closed orbits in a geometric setting.
- Finite-dimensional Nichols algebras are rigid objects in the category.
- Non-rigid Nichols algebras can be deformed into isomorphic pre- or post-Nichols algebras.

## Abstract

We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided fusion category $\mathcal{C}$. We describe such Hopf algebras as orbits for the action of a reductive group on an affine variety. We then show that the closed orbits are precisely the orbits of Nichols algebras, and that all other algebras are therefore deformations of Nichols algebras. For the case where the category $\mathcal{C}$ is the category $^G_G\mathcal{YD}$ of Yetter-Drinfeld modules over a finite group $G$, this reduces the question of generation by group-like elements and skew-primitives to a geometric question about rigidity of orbits. Comparing the results of Angiono Kochetov and Mastnak, this gives a new proof for the generation of finite dimensional pointed Hopf algebras with abelian groups of group-like elements by skew-primitives and group-like elements. We show that if $V$ is a simple object in $\mathcal{C}$ and $\text{B}(V)$ is finite dimensional, then $\text{B}(V)$ must be rigid. We also show that a non-rigid Nichols algebra can always be deformed to a pre-Nichols algebra or a post-Nichols algebra which is isomorphic to the Nichols algebra as an object of the category $\mathcal{C}$.