Finite GK-dimensional Nichols Algebras over the Infinite Dihedral Group

TL;DR

This paper classifies Nichols algebras over the infinite dihedral group, identifying which have finite Gelfand-Kirillov dimension by analyzing irreducible Yetter-Drinfeld modules.

Contribution

It provides a complete classification of irreducible Yetter-Drinfeld modules over the infinite dihedral group and determines which Nichols algebras have finite GK-dimension.

Findings

Identified all irreducible Yetter-Drinfeld modules over $ ext{D}_ ext{infty}$

Determined conditions for Nichols algebras to have finite GK-dimension

Contributed to the classification of Hopf algebras with finite GK-dimension

Abstract

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over $\mathbb{D}_{\infty}$, the infinite dihedral group. We find all the irreducible Yetter-Drinfeld modules $V$ over $\mathbb{D}_{\infty}$, and determine which Nichols algebras $\mathcal{B}(V)$ of $V$ are finite GK-dimensional.