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 ε A ∑S-1(a<1>)a<-1> a},<0> = 1H ⊗ a}, the subalgebra of coinvariants of the Verma structure A ε 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 ⊗B A → H ⊗ A, β(a ⊗B b) = ∑S-1(b<-1>) ⊗ ab<0> is surjective (i.e., A/B is a quantum homogeneous space), then the induction fu...
Integrals, quantum Galois extensions and the affineness criterion for quantum Yetter-Drinfel'd modules
MENINI, Claudia;
2002
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 ε A ∑S-1(a<1>)a<-1> a},<0> = 1H ⊗ a}, the subalgebra of coinvariants of the Verma structure A ε 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 ⊗B A → H ⊗ A, β(a ⊗B b) = ∑S-1(b<-1>) ⊗ ab<0> is surjective (i.e., A/B is a quantum homogeneous space), then the induction fu...I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
