Integrals, Quantum Galois Extensions, and the Affineness Criterion for Quantum Yetter–Drinfel'd Modules

Abstract

In this paper we shall generalize the notion of an integral on a Hopf algebra introduced by Sweedler, by defining the more general concept of an integral of a threetuple (H,A,C), where H is a Hopf algebra coacting on an algebra A and acting on a coalgebra C. We prove that there exists a total integral γ:C→Hom(C,A) of (H,A,C) if and only if any representation of (H,A,C) is injective in a functorial way, as a corepresentation of C. In particular, the quantum integrals associated to Yetter–Drinfel'd modules are defined. Let now A be an H-bicomodule algebra, HYDA the category of quantum Yetter–Drinfel'd modules, and B={a∈A|∑S−1(a〈1〉)a〈−1〉⊗a〈0〉=1H⊗a}, the subalgebra of coinvariants of the Verma structure A∈HYDA. We shall prove the following affineness criterion: if there exists γ:H→Hom(H,A) a total quantum integral and the canonical map β:A⊗BA→H⊗A, β(a⊗Bb)=∑S−1(b〈1〉)b〈−1〉⊗ab〈0〉 is surjective (i.e., A/B is a quantum homogeneous space), then the induction functor –⊗BA:MB→HYDA is an equivalence of categories. The affineness criteria proven by Cline, Parshall, and Scott, and independently by Oberst (for affine algebraic groups schemes) and Schneider (in the noncommutative case), are recovered as special cases.