Quantum Integrals And The Affineness Criterion For Quantum Yetter-Drinfeld π-Modules

In the paper, the quantum integrals associated to quantum Yetter-Drinfeld pi-modules are defined. We shall prove the following affineness criterion: if there exists theta = {theta(beta) : H-beta -> Hom(H beta-1, A)}(beta is an element of pi), a total quantum integral and the canonical map chi : A circle times(B) A -> circle plus(gamma is an element of pi) H-gamma circle times A, chi(a circle times(B) b) = circle plus(gamma is an element of pi) S-gamma(-1) phi(alpha)(b([1,alpha-1 gamma-1 alpha]))b([0,0]<-1,gamma >) circle times ab([0,0]< 0,0 >) is surjective. Then the induction functor - circle times(B) A : u(B) ->(H) yD(Lambda)(alpha) is an equivalence of categories. The affineness criterion proven by Menini and Militaru is recovered as special cases.