A correspondence theorem for modules over Hopf algebras

Abstract:

Let H be a finite-dimensional Hopf algebra. We prove that if M is a faithful H-module and if ${H_1} \ne {H_2}$ are sub-Hopf algebras of H, then ${M^{{H_1}}} \ne {M^{{H_2}}}$, where ${M^{{H_1}}}$ and ${M^{{H_2}}}$ are the invariants in M under the respective actions of ${H_1}$ and ${H_2}$. We also show that if ${H_1} \ne {H_2}$, then ${H_1}$ and ${H_2}$ have different left integrals. Both of these results rely heavily on the freeness theorem of Nichols-Zoeller.