Irreducible actions and faithful actions of Hopf algebras (Q1173842)

The authors investigate the structure of an algebra \(A\) with a Hopf algebra \(H\) acting on it. \(A\) is considered as a module over the smash product \(A\# H\). If \(A\) is of finite Goldie rank and irreducible as an \(A\# H\)-module then the (right-) dimension of \(A\) over the fix ring \(A^ H\), a division ring, is bounded above by the dimension of \(H\). Various applications and open questions about group actions and actions by derivations are discussed. In particular for a division ring \(D\) the inequality \([D: D^ H]\leq \dim H\) holds. A nice side-result is the existence of polynomial identities for certain prime rings \(A\) with a central ring of invariants. Furthermore it is proved that \(A\# H\) is prime iff the non-zero (left/right) \(A\# H\)-submodules of \(A\) are faithful. Many illustrating examples for the theorems are given.