Hopf quivers and Nichols algebras in positive characteristic

TL;DR

This paper develops a combinatorial approach to analyze Nichols algebras in positive characteristic, leading to new explicit examples and the construction of novel finite-dimensional pointed Hopf algebras.

Contribution

It introduces a combinatorial formula for products in Hopf quiver algebras and applies it to construct new Nichols algebras and associated Hopf algebras in positive characteristic.

Findings

Explicit new examples of Nichols algebras in positive characteristic

Descriptions of Radford biproducts and liftings of these biproducts

Construction of new finite-dimensional pointed Hopf algebras

Abstract

We apply a combinatorial formula of the first author and Rosso, for products in Hopf quiver algebras, to determine the structure of Nichols algebras. We illustrate this technique by explicitly constructing new examples of Nichols algebras in positive characteristic. We further describe the corresponding Radford biproducts and some liftings of these biproducts, which are new finite dimensional pointed Hopf algebras.