## Yetter-Drinfeld modules over weak Hopf algebras and the center construction, arXiv:math.QA/0409599

Citations: | 2 - 0 self |

### BibTeX

@MISC{Caenepeel_yetter-drinfeldmodules,

author = {S. Caenepeel and Dingguo Wang and Yanmin Yin},

title = {Yetter-Drinfeld modules over weak Hopf algebras and the center construction, arXiv:math.QA/0409599},

year = {}

}

### OpenURL

### Abstract

H, and show that the category of Yetter-Drinfeld modules is isomorphic to

### Citations

62 |
Weak Hopf algebras
- Böhm, Nill, et al.
- 1999
(Show Context)
Citation Context ...-coaction given by (78) and (79) is also a left-left Yetter-Drinfeld module. Proof. We have to show that λ(h · m ∗ ) = ∑ i 〈m∗, S(h (1))ni[0]〉S −1 (ni[−1]) ⊗ ni is equal to h (1)m ∗ [−1] S(h (3)) ⊗ h =-=[2]-=-m ∗ ∑ [−1] = i 〈m∗, ni[0]〉h (1)S −1 (ni[−1])S(h(3)) ⊗ (h(2) · n ∗ i ). It suffices to show that both terms coincide after we evaluate the second tensor factor at an arbitrary m ∈ M. Indeed, ∑ i 〈m∗ , ... |

36 |
Unifying Hopf modules
- Doi
- 1992
(Show Context)
Citation Context ...], in the case of quasi-Hopf algebras. In [6], it was observed that Yetter-Drinfeld modules over a classical Hopf algebra are special cases of Doi-Hopf modules, as introduced by Doi and Koppinen (see =-=[7, 11]-=-). In Section 3, we will show that Yetter-Drinfeld modules over weak Hopf algebras are weak Doi-Hopf modules, in the sense of Böhm [1], and, a fortiori, weak entwined modules [5], and comodules over a... |

33 |
Variations on the smash product with applications to group-graded rings
- Koppinen
- 1995
(Show Context)
Citation Context ...], in the case of quasi-Hopf algebras. In [6], it was observed that Yetter-Drinfeld modules over a classical Hopf algebra are special cases of Doi-Hopf modules, as introduced by Doi and Koppinen (see =-=[7, 11]-=-). In Section 3, we will show that Yetter-Drinfeld modules over weak Hopf algebras are weak Doi-Hopf modules, in the sense of Böhm [1], and, a fortiori, weak entwined modules [5], and comodules over a... |

31 | Finite quantum groupoids and their applications - Nikshych, Vainerman - 2002 |

30 |
Tortile Yang-Baxter operators in tensor categories
- Joyal, Street
- 1991
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Citation Context ... modules over weak Hopf algebras. To this end, we proceed as follows. The category of modules over a weak Hopf algebra is a monoidal category. The center of a monoidal category has been introduced in =-=[9]-=- and [12]; the center construction provides a tool to turn monoidal categories into braided monoidal categories. For example, the center of the category of G-sets is the category of Whitehead crossed ... |

26 |
Representations, duals and quantum doubles of monoidal categories
- Majid
- 1991
(Show Context)
Citation Context ... over weak Hopf algebras. To this end, we proceed as follows. The category of modules over a weak Hopf algebra is a monoidal category. The center of a monoidal category has been introduced in [9] and =-=[12]-=-; the center construction provides a tool to turn monoidal categories into braided monoidal categories. For example, the center of the category of G-sets is the category of Whitehead crossed G-sets, a... |

19 |
Duality for Generalized Kac Algebras and a Characterization of Finite Groupoid Algebras
- Yamanouchi
- 1994
(Show Context)
Citation Context ...ity of the unit are replaced by weaker axioms. The easiest example of a weak Hopf algebra is a groupoid algebra; other examples are face algebras [8], quantum groupoids [15], generalized Kac algebras =-=[17]-=- and quantum transformation groupoids [14]. Temperley-Lieb algebras give rise to weak Hopf algebras (see [14]). A purely algebraic study of weak Hopf algebras has been presented in [2]. A survey of we... |

17 | Doi-Koppinen modules over weak Hopf algebras
- Bohm
(Show Context)
Citation Context ...s of Doi-Hopf modules, as introduced by Doi and Koppinen (see [7, 11]). In Section 3, we will show that Yetter-Drinfeld modules over weak Hopf algebras are weak Doi-Hopf modules, in the sense of Böhm =-=[1]-=-, and, a fortiori, weak entwined modules [5], and comodules over a coring ([3]). The advantage of this approach is that it leads easily to the description of the Drinfeld double, in the case of a fini... |

17 |
Modules over weak entwining structures
- Caenepeel, Groot
(Show Context)
Citation Context ...and Koppinen (see [7, 11]). In Section 3, we will show that Yetter-Drinfeld modules over weak Hopf algebras are weak Doi-Hopf modules, in the sense of Böhm [1], and, a fortiori, weak entwined modules =-=[5]-=-, and comodules over a coring ([3]). The advantage of this approach is that it leads easily to the description of the Drinfeld double, in the case of a finite dimensional weak Hopf algebra (or, more g... |

16 |
Quantum group symmetry of partition functions of IRF models and its application to Jones’ index theory
- Hayashi
- 1993
(Show Context)
Citation Context ...the multiplicativity of the counit and comultiplicativity of the unit are replaced by weaker axioms. The easiest example of a weak Hopf algebra is a groupoid algebra; other examples are face algebras =-=[8]-=-, quantum groupoids [15], generalized Kac algebras [17] and quantum transformation groupoids [14]. Temperley-Lieb algebras give rise to weak Hopf algebras (see [14]). A purely algebraic study of weak ... |

13 |
Crossed modules and Doi-Hopf modules
- Caenepeel, Militaru, et al.
- 1997
(Show Context)
Citation Context ...MIN YIN left-right, right-left and right-right versions, are isomorphic as braided monoidal categories. Here we apply methods that have been used before in [4], in the case of quasi-Hopf algebras. In =-=[6]-=-, it was observed that Yetter-Drinfeld modules over a classical Hopf algebra are special cases of Doi-Hopf modules, as introduced by Doi and Koppinen (see [7, 11]). In Section 3, we will show that Yet... |

13 |
Quantum Cohomology, Quantum Groupoids, and Subfactors, unpublished talk presented at the First Caribic
- Ocneanu
(Show Context)
Citation Context ... the counit and comultiplicativity of the unit are replaced by weaker axioms. The easiest example of a weak Hopf algebra is a groupoid algebra; other examples are face algebras [8], quantum groupoids =-=[15]-=-, generalized Kac algebras [17] and quantum transformation groupoids [14]. Temperley-Lieb algebras give rise to weak Hopf algebras (see [14]). A purely algebraic study of weak Hopf algebras has been p... |

12 | Invariants of knots and 3-manifolds from quantum groupoids, Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of Three-Manifolds
- Nikshych, Turaev, et al.
- 1999
(Show Context)
Citation Context ... commutative ring), using methods developed in [5]. This is what we will do in Section 4. We find a version of the Drinfeld double, and show that it is isomorphic to the Drinfeld double introduced in =-=[13]-=-. In Sections 1.1 and 1.2, we recall some general properties of weak bialgebras and Hopf algebras. Further detail can be found in [3, 2, 14] In Section 1.3, we recall the center construction, and in S... |

9 | Weak Hopf algebras and quantum groupoids
- Schauenburg
- 2003
(Show Context)
Citation Context ...The source and target space are anti-isomorphic, and they are separable Frobenius algebras over k. This was first proved for weak Hopf algebras (see [2]), and then generalized to weak bialgebras (see =-=[16]-=-). Lemma 1.2. [16] Let H be a weak bialgebra. Then εs restricts to an anti-algebra isomorphism Ht → Hs with inverse εt, and εt restricts to an anti-algebra isomorphism Hs → Ht with inverse εs. Proposi... |

7 | Yetter-Drinfeld categories for quasi-Hopf algebras
- Bulacu, Caenepeel, et al.
(Show Context)
Citation Context ...nts. 12 S. CAENEPEEL, DINGGUO WANG, AND YANMIN YIN left-right, right-left and right-right versions, are isomorphic as braided monoidal categories. Here we apply methods that have been used before in =-=[4]-=-, in the case of quasi-Hopf algebras. In [6], it was observed that Yetter-Drinfeld modules over a classical Hopf algebra are special cases of Doi-Hopf modules, as introduced by Doi and Koppinen (see [... |