Relative Hom-Hopf modules and total integrals

Abstract

Let $(H, \a)$ be a monoidal Hom-Hopf algebra and $(A, \b)$ a right $(H, \a)$-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of $(A, \b)$ in the setting of monoidal Hom-Hopf algebras. Also we prove that there exists a total integral $ϕ: (H, \a)\rightarrow (A, \b)$ if and only if any representation of the pair $(H,A)$ is injective in a functorial way, as a corepresentation of $(H, \a)$, which generalizes Doi's result. Finally, we define a total quantum integral $\g: H\rightarrow Hom(H, A)$ and prove the following affineness criterion: if there exists a total quantum integral $\g$ and the canonical map $ψ: Aø_{B}A\rightarrow AøH,\ \ aø_{B}b\mapsto \b^{-1}(a)b_{[0]}ø\a(b_{[1]}) $is surjective, then the induction functor $A{ø}_B-:\tilde{\mathcal{H}}({\mathcal{M}}_k{)}_B\to \tilde{\mathcal{H}}({\mathcal{M}}_k{)}_A^H$ is an equivalence of categories.

20 pages. arXiv admin note: text overlap with arXiv:math/0106067 by other authors

References in corpus (2)

• Hom-algebras and homology
• Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras