Pointed Hopf Algebras with classical Weyl Groups

TL;DR

This paper classifies when Nichols algebras over classical Weyl groups are finite or infinite dimensional, providing key conditions and exceptions, which advances understanding of pointed Hopf algebras related to these groups.

Contribution

It establishes necessary and sufficient conditions for the finiteness of Nichols algebras over classical Weyl groups, identifying three specific cases of exception.

Findings

Most Nichols algebras over classical Weyl groups are infinite dimensional.

Finite dimensional Nichols algebras occur only in three specific cases.

Provides a complete characterization of finiteness conditions for these algebras.

Abstract

We prove that Nichols algebras of irreducible Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $\mathbb S_n$ are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $A$ to be finite dimensional.