Nichols algebras over classical Weyl groups (II)

TL;DR

This paper classifies conjugacy classes in classical Weyl groups and demonstrates that Nichols algebras over these groups are generally infinite dimensional, with specific exceptions identified.

Contribution

It provides a classification of conjugacy classes of classical Weyl groups and determines when Nichols algebras are finite or infinite dimensional.

Findings

Most conjugacy classes are of type D, leading to infinite-dimensional Nichols algebras.

Finite-dimensional Nichols algebras occur only in specific classes in S_5 and S_n.

The work extends understanding of Nichols algebras over classical Weyl groups.

Abstract

It is shown that except in three cases conjugacy classes of classical Weyl groups $W(B_{n})$ and $W(D_{n})$ are of type ${\rm D}$. This proves that Nichols algebras of irreducible Yetter-Drinfeld modules over the classical Weyl groups $\mathbb W_{n}$ (i.e. $H_{n}\rtimes \mathbb{S}_{n}$) are infinite dimensional, except the class of type $(2, 3),(1^{2}, 3)$ in $\mathbb S_{5}$, and $(1^{n-2}, 2)$ in $\mathbb S_{n}$ for $n >5$.