### Abstract

This chapter gives a motivation for the use of Hopf algebras and their various generalizations for describing symmetries in diverse situations. This is made possible by their common feature of a closed monoidal structure of the category of modules—lifted from different base categories in each case of the discussed Hopf algebra-like structures. In category theoretical terms, liftings of (closed) monoidal structures to module categories correspond to so-called (Hopf) bimonads. The aim of the book, as summarized in the Introduction, is to give a common interpretation of the apparently different generalizations of Hopf algebra as Hopf monad structures on suitable functors. The covered examples include classical Hopf algebras, Hopf monoids in duoidal (in particular braided monoidal) categories, Hopf algebroids and (in particular) weak Hopf algebras.

Original language | English |
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Pages (from-to) | 1-6 |

Number of pages | 6 |

Journal | Lecture Notes in Mathematics |

Volume | 2226 |

DOIs | |

Publication status | Published - Jan 1 2018 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Lecture Notes in Mathematics*,

*2226*, 1-6. https://doi.org/10.1007/978-3-319-98137-6_1

Research output: Contribution to journal › Editorial

*Lecture Notes in Mathematics*, vol. 2226, pp. 1-6. https://doi.org/10.1007/978-3-319-98137-6_1

}

TY - JOUR

T1 - Introduction

AU - Böhm, G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - This chapter gives a motivation for the use of Hopf algebras and their various generalizations for describing symmetries in diverse situations. This is made possible by their common feature of a closed monoidal structure of the category of modules—lifted from different base categories in each case of the discussed Hopf algebra-like structures. In category theoretical terms, liftings of (closed) monoidal structures to module categories correspond to so-called (Hopf) bimonads. The aim of the book, as summarized in the Introduction, is to give a common interpretation of the apparently different generalizations of Hopf algebra as Hopf monad structures on suitable functors. The covered examples include classical Hopf algebras, Hopf monoids in duoidal (in particular braided monoidal) categories, Hopf algebroids and (in particular) weak Hopf algebras.

AB - This chapter gives a motivation for the use of Hopf algebras and their various generalizations for describing symmetries in diverse situations. This is made possible by their common feature of a closed monoidal structure of the category of modules—lifted from different base categories in each case of the discussed Hopf algebra-like structures. In category theoretical terms, liftings of (closed) monoidal structures to module categories correspond to so-called (Hopf) bimonads. The aim of the book, as summarized in the Introduction, is to give a common interpretation of the apparently different generalizations of Hopf algebra as Hopf monad structures on suitable functors. The covered examples include classical Hopf algebras, Hopf monoids in duoidal (in particular braided monoidal) categories, Hopf algebroids and (in particular) weak Hopf algebras.

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UR - http://www.scopus.com/inward/citedby.url?scp=85056255534&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-98137-6_1

DO - 10.1007/978-3-319-98137-6_1

M3 - Editorial

VL - 2226

SP - 1

EP - 6

JO - Lecture Notes in Mathematics

T2 - Lecture Notes in Mathematics

JF - Lecture Notes in Mathematics

SN - 0075-8434

ER -