Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups

TL;DR

This paper classifies finite-dimensional Nichols algebras over solvable groups in characteristic zero, showing all such algebras over odd order groups are diagonal type, leading to a complete description of pointed Hopf algebras of odd dimension.

Contribution

It provides a comprehensive classification of Nichols algebras over solvable groups and characterizes pointed Hopf algebras of odd dimension as diagonal type.

Findings

All Nichols algebras over groups of odd order are of diagonal type.

Complete classification of pointed Hopf algebras of odd dimension.

Reduction to positive characteristic as a key method.

Abstract

We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a consequence, all Nichols algebras over groups of odd order are of diagonal type, which allows us to describe all pointed Hopf algebras of odd dimension.