On Nichols algebras over PGL(2,q) and PSL(2,q)

Abstract

We compute necessary conditions on Yetter-Drinfeld modules over the groups $\mathbf{PGL}(2,q)=\mathbf{PGL}(2,\FF_q)$ and $\mathbf{PSL}(2,q)=\mathbf{PSL}(2,\FF_q)$ to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that there is no non-trivial finite-dimensional pointed Hopf algebra over the Mathieu groups $M_{20}$ and $M_{21}=\mathbf{PSL}(3,4)$.

Minor changes

References in corpus (2)

• On pointed Hopf algebras associated to unmixed conjugacy classes in S_n
• On Nichols algebras over SL(2,Fq) and GL(2,Fq)

Cited by in corpus (9)

• Finite-dimensional pointed Hopf algebras with alternating groups are trivial
• Pointed Hopf algebras over the sporadic simple groups
• On pointed Hopf algebras over dihedral groups
• Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSL(n,q)
• On Hopf Algebras over quantum subgroups
• Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$
• Pointed Hopf algebras over some sporadic simple groups
• Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL(n,q)
• Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups