Pointed Hopf algebras over the sporadic simple groups

TL;DR

This paper classifies finite-dimensional complex pointed Hopf algebras over sporadic simple groups, showing most are group algebras except for three cases where the Nichols algebra's finiteness is unresolved.

Contribution

It proves that all such Hopf algebras are group algebras except for three sporadic groups, providing a list of modules with unknown Nichols algebra finiteness.

Findings

Most sporadic group-based pointed Hopf algebras are group algebras.

Identifies three sporadic groups with unresolved Nichols algebra finiteness.

Provides a list of modules for the exceptional groups.

Abstract

We show that every finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group is a group algebra, except for the Fischer group Fi22, the Baby Monster and the Monster. For these three groups, we give a short list of irreducible Yetter-Drinfeld modules whose Nichols algebra is not known to be finite-dimensional.