Finite-dimensionality of Nichols algebras for simple Yetter–Drinfeld modules over non-abelian groups

Determine necessary and sufficient conditions under which the Nichols algebra B(M(O,ρ)) of a simple Yetter–Drinfeld module M(O,ρ) over kG, constructed from a conjugacy class O in a non-abelian finite group G and an irreducible representation ρ of the centralizer G^g, is finite-dimensional.

Background

For non-abelian groups, simple Yetter–Drinfeld modules are parametrized by pairs (O,ρ), where O is a conjugacy class and ρ is an irreducible representation of the corresponding centralizer. While the finite-dimensionality problem has been resolved in several multi-summand cases via Weyl groupoids, the single-summand case remains unresolved and is central to understanding Nichols algebras over non-abelian groups.