The affineness criterion for quantum Hom-Yetter–Drinfel’d modules

Shuangjian Guo¹ ; Shengxiang Wang²

¹ School of Mathematics and Statistics Guizhou University of Finance and Economics Guiyang, 550025, P.R. China
² School of Mathematics and Finance Chuzhou University Chuzhou 239000, P.R. China

Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 169-185

Résumé

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.

Détail

DOI : 10.4064/cm6609-12-2015

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