### Abstract

In this chapter, the R-bialgebroids and Hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra R carries a so-called separable Frobenius structure. Separable Frobenius structures on some algebra R are shown to correspond to separable Frobenius structures on the forgetful functor from the category of R-bimodules to the category of vector spaces. Based on that, the bijection of Chap. 5 is refined to bijections between three structures, for any algebra A. First, monoidal structures on the category of A-modules together with separable Frobenius structures on the forgetful functor to the category of vector spaces. Second, bialgebroid structures on A over some base algebra R, together with separable Frobenius structures on R. Finally, weak bialgebra structures on A. The R-bialgebroid A is a Hopf algebroid if and only if the corresponding weak bialgebra A is a weak Hopf algebra.

Original language | English |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 75-97 |

Number of pages | 23 |

DOIs | |

Publication status | Published - Jan 1 2018 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2226 |

ISSN (Print) | 0075-8434 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Lecture Notes in Mathematics*(pp. 75-97). (Lecture Notes in Mathematics; Vol. 2226). Springer Verlag. https://doi.org/10.1007/978-3-319-98137-6_6

**Weak (Hopf) bialgebras.** / Böhm, G.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Lecture Notes in Mathematics.*Lecture Notes in Mathematics, vol. 2226, Springer Verlag, pp. 75-97. https://doi.org/10.1007/978-3-319-98137-6_6

}

TY - CHAP

T1 - Weak (Hopf) bialgebras

AU - Böhm, G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In this chapter, the R-bialgebroids and Hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra R carries a so-called separable Frobenius structure. Separable Frobenius structures on some algebra R are shown to correspond to separable Frobenius structures on the forgetful functor from the category of R-bimodules to the category of vector spaces. Based on that, the bijection of Chap. 5 is refined to bijections between three structures, for any algebra A. First, monoidal structures on the category of A-modules together with separable Frobenius structures on the forgetful functor to the category of vector spaces. Second, bialgebroid structures on A over some base algebra R, together with separable Frobenius structures on R. Finally, weak bialgebra structures on A. The R-bialgebroid A is a Hopf algebroid if and only if the corresponding weak bialgebra A is a weak Hopf algebra.

AB - In this chapter, the R-bialgebroids and Hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra R carries a so-called separable Frobenius structure. Separable Frobenius structures on some algebra R are shown to correspond to separable Frobenius structures on the forgetful functor from the category of R-bimodules to the category of vector spaces. Based on that, the bijection of Chap. 5 is refined to bijections between three structures, for any algebra A. First, monoidal structures on the category of A-modules together with separable Frobenius structures on the forgetful functor to the category of vector spaces. Second, bialgebroid structures on A over some base algebra R, together with separable Frobenius structures on R. Finally, weak bialgebra structures on A. The R-bialgebroid A is a Hopf algebroid if and only if the corresponding weak bialgebra A is a weak Hopf algebra.

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BT - Lecture Notes in Mathematics

PB - Springer Verlag

ER -