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Journals and Conferences
A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that… (More)
Let H be a finite dimensional quasi-Hopf algebra over a field k and A a right H-comodule algebra in the sense of . We first show that on the k-vector space A⊗H∗ we can define an algebra structure, denoted by A # H∗, in the monoidal category of left H-modules (i.e. A # H∗ is an Hmodule algebra in the sense of ). Then we will prove that the category of… (More)
We compute the representation-theoretic rank of a finite dimensional quasi-Hopf algebra H and of its quantum double D(H), within the rigid braided category of finite dimensional left D(H)-modules.
We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double D(H) of any finite dimensional quasi-Hopf algebra H is factorizable, and we characterize D(H) when H itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular,… (More)
We extend to the co-Frobenius case a result of Drinfeld and Radford related to the fourth power of the antipode of a finite dimensional (co) quasitriangular Hopf algebra.
Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in  and . In this note, we first generalize a result of Majid  for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of .
We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic. We prove also that the category H H YD fd of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute explicitly the canonical isomorphisms in H H YD fd . Finally, we show that certain duals of H0, the braided Hopf algebra… (More)
We generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional YetterDrinfeld module is again a Yetter-Drinfeld module. The algebra H0 in the category of Yetter-Drinfeld modules that can be obtained by modifying the multiplication in a proper way is quantum commutative. We give a… (More)
We introduce and investigate the basic properties of an involutory (dual) quasi-Hopf algebra. We also study the representations of an involutory quasi-Hopf algebra and prove that an involutory dual quasi-Hopf algebra with non-zero integral is cosemisimple.