Clifford correspondence for finite dimensional Hopf algebras (Q1306905)

Classical Clifford theory studies the relationship between the representations of a group and the representations of a normal subgroup. Let \(H\) be a finite dimensional Hopf algebra over \(k\) and let \(A=B\#_\sigma H\) be a crossed product algebra of a \(k\)-algebra \(B\) with \(H\). In the present paper, the author generalizes the Clifford correspondence for the aforementioned extension. It is shown that if \(T\) is an irreducible \(A\)-stable \(B\)-module, there exists an equivalence between the category of finite dimensional \(A\)-modules whose restriction of \(B\) is isomorphic to a direct sum of copies of \(T\) and the category of finite dimensional \(E\)-modules, where \(E\) is the opposite endomorphism ring of the induced \(A\)-module, \(A\otimes_BT\). Assuming that \(H\) is cocommutative, a one-to-one correspondence in the general case, when \(T\) is not necessarily \(A\)-stable, is also given.