Finiteness of the number of coideal subalgebras

Abstract

It is proved that any finite dimensional Hopf algebra which is either semisimple or cosemisimple has finitely many right coideal subalgebras. As a consequence, over an algebraically closed base field any action of a finite dimensional cosemisimple Hopf algebra on a commutative domain factors through an action of a group algebra. This extends two results of Etingof and Walton to the case where the Hopf algebra is cosemisimple, but not necessarily semisimple.

Subrings of invariants for actions of finite dimensional Hopf algebras

“…The next result has been obtained by Etingof and Walton [25] when either char k = 0 or char k > 0 and H is also semisimple. Its extension to the case when H is not necessarily semisimple has been given in [53].…”

Affine commutative-by-finite Hopf algebras

“…Thus this left adjoint action on D ′ factors through H ′ A ′+ = A ′+ H ′ . Since H ′ /A ′+ H ′ is semisimple and cosemisimple and D ′ is a commutative domain by (9), this action in turn factors through an inner faithful group action by [67,Theorem 2], say H ′ /I ∼ = kΛ for some finite group Λ and some Hopf ideal I of H ′ that annihilates D ′ under the left adjoint action. However, by (4)(iii) the left adjoint action of H on A ′ factors through D + H with H/D + H ∼ = kΓ acting inner faithfully.…”

“…In §7 we focus on the case where the finite dimensional Hopf factor H can be chosen to be semisimple and cosemisimple. Results of Etingof, Walton and Skryabin [25,67] are crucial to the key message about this class of algebras, which is that the underlying noncommutativity is generated by the action of a finite group. Theorem 7.2, the main result of §7, is rather complex to state; the following is the special case where H is assumed to be prime, avoiding many of the technicalities.…”