Lecture notes on Nichols algebras

TL;DR

This paper provides an accessible introduction to Nichols algebras, emphasizing categorical perspectives, examples, and their role in constructing quantum groups and tensor categories, with applications in conformal field theory.

Contribution

It offers a new categorical approach to Nichols algebras, making the topic more accessible and connecting it to quantum groups and conformal field theory.

Findings

Explanation of Nichols algebra properties and reflection theory

Construction of quantum groups from Nichols algebras

Examples beyond diagonal cases and applications in CFT

Abstract

These are lecture notes for an introductory course on Nichols algebras. As a main reference, I work with the book by Heckenberger and Schneider, but I want to take a distinct categorical perspective and try to develop the topic for an audience without a background in Hopf algebras. On the other hand I put some emphasis on hands-on examples. My first goal is to explain the definitions and the striking properties of Nichols algebras, foremost the odd reflection theory that is already present in Lie superalgebras. My second goal is to explain how the category of representations of a quantum group can be constructed, using categorical tools, from the Nichols algebra as its centerpiece. This makes the zoo of different existing versions of quantum groups more transparent and allows the construction of many more non-semisimple modular tensor categories. Other topics include different types of examples beyond the diagonal case, categorical versions of some Hopf algebra constructions, and an outlook section on the appearance of Nichols algebras in conformal field theory.