Finite dimensional Nichols algebras over finite cyclic groups

TL;DR

This paper classifies finite dimensional Nichols algebras over finite cyclic groups, showing that over Z_2 they are quantum linear spaces, while for larger modules they are infinite dimensional.

Contribution

It provides a complete classification of finite dimensional Nichols algebras over finite cyclic groups, identifying conditions for finiteness and structure.

Findings

Finite dimensional Nichols algebras over Z_2 are quantum linear spaces.

Nichols algebras over Z_n with dim V > 3 are infinite dimensional.

Complete classification of Nichols algebras over finite cyclic groups.

Abstract

All finite dimensional Nichols algebras with diagonal type of connected finite dimensional Yetter-Drinfeld modules over finite cyclic group $\mathbb Z_n$ are found. It is proved that finite dimensional Nichols algebra over $\mathbb Z_2$ is a quantum linear space and Nichols algebra of connected Yetter-Drinfeld module $V$ over $\mathbb Z_n$ with $\dim V >3$ is infinite dimensional.