The affineness criterion for quantum Hom-Yetter–Drinfel’d modules
Volume 143 / 2016
Abstract
Quantum integrals associated to quantum Hom-Yetter–Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter–Drinfel’d modules is proved in the following form. Let $(H, \alpha)$ be a monoidal Hom-Hopf algebra, $(A, \beta)$ an $(H, \alpha)$-Hom-bicomodule algebra and $B=A^{\mathop{\rm co}H}$. Under the assumption that there exists a total quantum integral $\gamma: H\rightarrow {\rm Hom}(H,A)$ and the canonical map $\beta: A\otimes_{B}A\rightarrow A\otimes H$, $a\otimes_{B}b\mapsto S^{-1}(b_{[1]})\alpha(b_{[0][-1]}) \otimes \beta^{-1}(a)\beta(b_{[0][0]})$, is surjective, we prove that the induction functor $A\otimes_B-:\widetilde{{\mathscr H}} ({\mathscr M}_k)_{B}\rightarrow {}^H{\mathscr H}{\mathscr Y}{\mathscr D}_A$ is an equivalence of categories.
