## Adjointable monoidal functors and quantum groupoids, Hopf algebras in noncommutative geometry and physics

Venue: | Lecture Notes in Pure and |

Citations: | 13 - 2 self |

### BibTeX

@INPROCEEDINGS{Szlachányi_adjointablemonoidal,

author = {Kornél Szlachányi},

title = {Adjointable monoidal functors and quantum groupoids, Hopf algebras in noncommutative geometry and physics},

booktitle = {Lecture Notes in Pure and},

year = {},

pages = {291--307}

}

### OpenURL

### Abstract

Abstract. Every monoidal functor G: C → M has a canonical factorization through the category RMR of bimodules in M over some monoid R in M in which the factor U: C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given by Schauenburg [15] together with their bimonad description given by the author in [18] here we characterize the ”long ” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1.

### Citations

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Categories for the Working Mathematician, 2nd edition
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- 1998
(Show Context)
Citation Context ...ained that way are precisely the monads which have right adjoints. For this characterization of algebras the closed monoidal structure of Mk is essential. The Eilenberg-Moore category of ”T-algebras” =-=[11, 2]-=-, or perhaps better to say, ”T-modules” is nothing but the category MA of right Amodules equipped with the forgetful functor MA → Mk. Similarly, we can consider monads on the closed monoidal category ... |

78 |
Hopf algebroids and quantum groupoids
- Lu
- 1996
(Show Context)
Citation Context ...18] here we characterize the ”long” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1. Introduction Takeuchi’s ×R-bialgebras [20] or, what is the same [5], Lu’s bialgebroids =-=[9]-=- provide far reaching generalizations of the notion of bialgebra. A bialgebroid A is, roughly speaking, a bialgebra over some non-commutative k-algebra R. With noncommutativity of R, however, a new ph... |

64 |
Weak Hopf Algebras I
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- 1999
(Show Context)
Citation Context ...oid has more structure than just a monadic essentially strong monoidal functor with right adjoint. It is equipped also with an opmonoidal structure. For concreteness let 〈A, ∆, ǫ〉 be a weak bialgebra =-=[4]-=- and let R be identified with the canonical right subalgebra AR = {1(1)ǫ(a1(2)) | a ∈ A}. Then A becomes a right bialgebroid over R with (6.38) (6.39) (6.40) (6.41) s: R → A, r ↦→ r t: R op → A, r ↦→ ... |

50 |
An introduction to Tannaka duality and quantum groups, from: “Category theory
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- 1990
(Show Context)
Citation Context ...her one by Hayashi [6] which prove Tannaka duality theorems for Hopf algebroids and for face algebras, respectively. In their approach, as in that of Saavedra-Rivano, Deligne, Ulbrich and others (see =-=[13, 7]-=-) small categories are equipped with strong monoidal functors to a (sometimes rigid) category of bimodules and the task is to find a universal factorization through the comodule category of a quantum ... |

28 |
Handbook of Categorical Algebra 2
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(Show Context)
Citation Context ... a functor leads to the construction of a bialgebroid with Tannaka duality. We let V denote either kM, Ab or Set and work with V-monoidal V-categories and V-functors between them that have V-adjoints =-=[3]-=-. Due to the fact that V is not only symmetric monoidal closed but its monoidal unit is a generator, we can work with the underlying ordinary categories and consider V-functors as a special class of o... |

25 | Monads on tensor categories
- Moerdijk
(Show Context)
Citation Context ...uctures on the monad T = ⊗Re A. In a recent paper [18] bialgebroids have been characterized as the bimonads on RMR the underlying functors of which have right adjoints. A bimonad, or opmonoidal monad =-=[12, 10]-=-, is a monad in the 2-category MonopCat of Based on a talk given at the conference ”Hopf Algebras in Noncommutative Geometry and Physics” organized by Stefaan Caenapeel and Fred van Oystaeyen, Brussel... |

25 |
Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules
- Schauenburg
- 1998
(Show Context)
Citation Context ...over some monoid R in M in which the factor U : C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given by Schauenburg =-=[15]-=- together with their bimonad description given by the author in [18] here we characterize the ”long” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1. Introduction Takeuchi’... |

23 | Finite quantum groupoids and inclusions of finite type
- Szlachányi
(Show Context)
Citation Context ...these two obey compatibility conditions that can be called a separable Frobenius structure on the forgetful functor. This characterization of weak bialgebra forgetful functors was already sketched in =-=[17]-=- calling them ”split monoidal” functors. 2. The canonical factorization of monoidal functors Let C be a monoidal category with monoidal product ✷ : C × C → C and unit object e ∈ C. Then we have cohere... |

16 |
Modules, comodules and cotensor products over Frobenius algebras
- Abrams
- 1999
(Show Context)
Citation Context ...positely oriented, diagrams. In addition, there are compatibility conditions between the monoidal and opmonoidal structures that are not of the bialgebra type but rather of the Frobenius algebra type =-=[1]-=-. Namely, (6.46) (GX,Y ⊗ GZ) ◦ aGX,GY,GZ ◦ (GX ⊗ G Y,Z ) = G X ✷ Y,Z ◦ GaX,Y,Z ◦ GX,Y ✷ Z (GX ⊗ GY,Z) ◦ a −1 GX,GY,GZ ◦ (GX,Y ⊗ GZ) = G X,Y ✷ Z ◦ Ga −1 X,Y,Z ◦ GX (6.47) ✷ Y,Z for all A-modules X,Y , ... |

15 |
A canonical Tannaka duality for finite semisimple tensor categories, preprint
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- 1999
(Show Context)
Citation Context ...ng Tannaka theory from strong monoidal to monoidal functors. As for the state of the art of the traditional method we have to mention the recent papers by Phùng Hò Hái [14] and another one by Hayashi =-=[6]-=- which prove Tannaka duality theorems for Hopf algebroids and for face algebras, respectively. In their approach, as in that of Saavedra-Rivano, Deligne, Ulbrich and others (see [13, 7]) small categor... |

14 | Bialgebroids, ×A-Bialgebras and Duality
- Brzezinski, Militaru
(Show Context)
Citation Context ...iven by the author in [18] here we characterize the ”long” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1. Introduction Takeuchi’s ×R-bialgebras [20] or, what is the same =-=[5]-=-, Lu’s bialgebroids [9] provide far reaching generalizations of the notion of bialgebra. A bialgebroid A is, roughly speaking, a bialgebra over some non-commutative k-algebra R. With noncommutativity ... |

12 | The monoidal Eilenberg-Moore construction and bialgebroids
- Szlachányi
- 2003
(Show Context)
Citation Context ...unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given by Schauenburg [15] together with their bimonad description given by the author in =-=[18]-=- here we characterize the ”long” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1. Introduction Takeuchi’s ×R-bialgebras [20] or, what is the same [5], Lu’s bialgebroids [9]... |

11 | Galois actions by finite quantum groupoids in ”Locally Compact Quantum Groups and Groupoids
- Szlachányi
(Show Context)
Citation Context ...DJOINTABLE MONOIDAL FUNCTORS AND QUANTUM GROUPOIDS 13 bialgebra A is a bialgebroid over a separable Frobenius algebra R together with a Frobenius structure 〈ǫ, ∑ i ei ⊗ fi〉 in which ∑ i eifi = 1 (see =-=[19]-=- for more details). We call 〈R, ǫ, ∑ i ei ⊗ fi〉 a separable Frobenius structure. Accordingly, the forgetful functor GA : MA → Mk of a bialgebroid has more structure than just a monadic essentially str... |

7 |
Groups of algebras over A⊗A
- Takeuchi
- 1977
(Show Context)
Citation Context ...heir bimonad description given by the author in [18] here we characterize the ”long” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1. Introduction Takeuchi’s ×R-bialgebras =-=[20]-=- or, what is the same [5], Lu’s bialgebroids [9] provide far reaching generalizations of the notion of bialgebra. A bialgebroid A is, roughly speaking, a bialgebra over some non-commutative k-algebra ... |

6 | Opmonoidal monads
- McCrudden
(Show Context)
Citation Context ...uctures on the monad T = ⊗Re A. In a recent paper [18] bialgebroids have been characterized as the bimonads on RMR the underlying functors of which have right adjoints. A bimonad, or opmonoidal monad =-=[12, 10]-=-, is a monad in the 2-category MonopCat of Based on a talk given at the conference ”Hopf Algebras in Noncommutative Geometry and Physics” organized by Stefaan Caenapeel and Fred van Oystaeyen, Brussel... |

3 |
Dual bialgebroids for depth 2 ring extensions
- Kadison, Szlachányi
(Show Context)
Citation Context ..., moreover, k is a field then a separable Frobenius algebra, i.e., an algebra having a separable Frobenius structure (also called an index one Frobenius algebra), is nothing but a separable k-algebra =-=[8]-=-. Lemma 6.4. Let 〈R, µ, η, σ, ψ〉 be a separable Fobenius structure in Mk. Then the monoidal forgetful functor of bimodules Γ = ΓR : RMR → M has the following extension to a separable Frobenius structu... |

3 |
Quantum groups and non-commutative geometry, http://www.mathematik.uni-muenchen.de/∼pareigis/Vorlesungen/02SS/QGandNCG.pdf
- Pareigis
(Show Context)
Citation Context ...her one by Hayashi [6] which prove Tannaka duality theorems for Hopf algebroids and for face algebras, respectively. In their approach, as in that of Saavedra-Rivano, Deligne, Ulbrich and others (see =-=[13, 7]-=-) small categories are equipped with strong monoidal functors to a (sometimes rigid) category of bimodules and the task is to find a universal factorization through the comodule category of a quantum ... |

1 |
Quantum Groups, book available from ftp.mpce.mq.edu.au
- Street
(Show Context)
Citation Context ...noid in M. (2) Let m and n be monoids in C. If 〈b, λ, ρ〉 is an m-n bimodule in C then the triple 〈Gb, Gλ ◦ Gm,b, Gρ ◦ Gb,n〉 is a Gm-Gn bimodule in M. Proof. (1) is well-known and can be found e.g. in =-=[16]-=-. (2) is also known to many authors although an explicit proof is difficult to find. Just to advertise the statement4 KORNÉL SZLACHÁNYI we compute here commutativity of the left and right actions: λ ... |