### Abstract

The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.

Original language | English |
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Pages (from-to) | 563-599 |

Number of pages | 37 |

Journal | Algebras and Representation Theory |

Volume | 8 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 2005 |

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### Keywords

- (Quasi-)Frobenius extension
- Hopf algebroid
- Integral
- Maschke theorem

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Integral theory for Hopf algebroids.** / Böhm, G.

Research output: Contribution to journal › Article

*Algebras and Representation Theory*, vol. 8, no. 4, pp. 563-599. https://doi.org/10.1007/s10468-005-8760-0

}

TY - JOUR

T1 - Integral theory for Hopf algebroids

AU - Böhm, G.

PY - 2005/10

Y1 - 2005/10

N2 - The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.

AB - The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.

KW - (Quasi-)Frobenius extension

KW - Hopf algebroid

KW - Integral

KW - Maschke theorem

UR - http://www.scopus.com/inward/record.url?scp=27744454455&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27744454455&partnerID=8YFLogxK

U2 - 10.1007/s10468-005-8760-0

DO - 10.1007/s10468-005-8760-0

M3 - Article

VL - 8

SP - 563

EP - 599

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

SN - 1386-923X

IS - 4

ER -