On pointed Hopf algebras associated with the Mathieu simple groups

TL;DR

This paper investigates the structure of pointed Hopf algebras related to Mathieu simple groups, establishing conditions for their Nichols algebras to be infinite-dimensional or have negative braiding, and classifying certain finite-dimensional cases.

Contribution

It proves that Nichols algebras over Mathieu groups are either infinite-dimensional or have negative braiding, and classifies finite-dimensional pointed Hopf algebras for M22 and M24.

Findings

Nichols algebra B(O_s,ρ) is infinite-dimensional or has negative braiding.

Group algebra of M22 and M24 are the only finite-dimensional pointed Hopf algebras with these groups.

Classification results for pointed Hopf algebras associated with Mathieu groups.

Abstract

Let G be a Mathieu simple group, s in G, O_s the conjugacy class of s and \rho an irreducible representation of the centralizer of s. We prove that either the Nichols algebra B(O_s,\rho) is infinite-dimensional or the braiding of the Yetter-Drinfeld module M(O_s, \rho) is negative. We also show that if G=M22 or M24, then the group algebra of G is the only (up to isomorphisms) finite-dimensional complex pointed Hopf algebra with group-likes isomorphic to G.