Let H be a Hopf algebra and A an H-simple right H-comodule algebra. It is shown that under certain hypotheses every (H,A)-Hopf module is either projective or free as an A-module and A is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its f...

Let H be a Hopf algebra with bijective antipode, α, β ∈ AutHopf (H) and M a finite dimensional (α, β)-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k,H), the Brauer group of H .

We study branching laws for a classical group G and a symmetric subgroup H . Our approach is through the branching algebra, the algebra of covariants for H in the regular functions on the natural torus bundle over the flag manifold for G. We give concrete descriptions of certain subalgebras of the branching algebra using classical invariant theory. In this context, it turns out that the ten cla...

If H is a quasi-Hopf algebra and B is a right H-comodule algebra such that there exists v : H → B a morphism of right H-comodule algebras, we prove that there exists a left Hmodule algebra A such that B ≃ A#H . The main difference comparing to the Hopf case is that, from the multiplication of B, which is associative, we have to obtain the multiplication of A, which in general is not; for this w...

Assuming the continuum hypothesis, we show that the Calkin algebra has 2 ℵ 1 outer automorphisms. Let H be a separable infinite dimensional Hilbert space, let L(H) be the algebra of bounded operators on H, let K(H) be the algebra of compact operators on H, and let Q = L(H)/K(H) be the Calkin algebra. A long-standing problem asks whether every automorphism of Q is inner, that is, of the form x →...