# Relative Hom-Hopf modules and total integrals

## Abstract

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of (A, β) in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral ϕ : (H, α) → (A, β) if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of (H, α), which generalizes Doi’s result. Finally, we define a total quantum integral γ : H → Hom(H, A) and prove the following affineness criterion: if there exists a total quantum integral γ and the canonical map ψ : A⊗{sub B}A → A ⊗ H, a⊗{sub B}b ↦ β{sup −1}(a) b{sub [0]} ⊗ α(b{sub [1]}) is surjective, then the induction functor A⊗{sub B}−:ℋ{sup ~}(ℳ{sub k}){sub B}→ℋ{sup ~}(ℳ{sub k}){sub A}{sup H} is an equivalence of categories.

- Authors:

- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025 (China)
- Department of Mathematics, Southeast University, Nanjing 210096 (China)
- School of Mathematics and Finance, Chuzhou University, Chuzhou 239000 (China)
- (China)

- Publication Date:

- OSTI Identifier:
- 22403093

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; INDUCTION; INTEGRALS; MATHEMATICAL OPERATORS; QUANTUM GROUPS

### Citation Formats

```
Guo, Shuangjian, Zhang, Xiaohui, Wang, Shengxiang, E-mail: wangsx-math@163.com, and Department of Mathematics, Nanjing University, Nanjing 210093.
```*Relative Hom-Hopf modules and total integrals*. United States: N. p., 2015.
Web. doi:10.1063/1.4906938.

```
Guo, Shuangjian, Zhang, Xiaohui, Wang, Shengxiang, E-mail: wangsx-math@163.com, & Department of Mathematics, Nanjing University, Nanjing 210093.
```*Relative Hom-Hopf modules and total integrals*. United States. doi:10.1063/1.4906938.

```
Guo, Shuangjian, Zhang, Xiaohui, Wang, Shengxiang, E-mail: wangsx-math@163.com, and Department of Mathematics, Nanjing University, Nanjing 210093. Sun .
"Relative Hom-Hopf modules and total integrals". United States.
doi:10.1063/1.4906938.
```

```
@article{osti_22403093,
```

title = {Relative Hom-Hopf modules and total integrals},

author = {Guo, Shuangjian and Zhang, Xiaohui and Wang, Shengxiang, E-mail: wangsx-math@163.com and Department of Mathematics, Nanjing University, Nanjing 210093},

abstractNote = {Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of (A, β) in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral ϕ : (H, α) → (A, β) if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of (H, α), which generalizes Doi’s result. Finally, we define a total quantum integral γ : H → Hom(H, A) and prove the following affineness criterion: if there exists a total quantum integral γ and the canonical map ψ : A⊗{sub B}A → A ⊗ H, a⊗{sub B}b ↦ β{sup −1}(a) b{sub [0]} ⊗ α(b{sub [1]}) is surjective, then the induction functor A⊗{sub B}−:ℋ{sup ~}(ℳ{sub k}){sub B}→ℋ{sup ~}(ℳ{sub k}){sub A}{sup H} is an equivalence of categories.},

doi = {10.1063/1.4906938},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 56,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}