56 research outputs found

    Two-sided (two-cosided) Hopf modules and Doi-Hopf modules for quasi-Hopf algebras

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    Let HH be a finite dimensional quasi-Hopf algebra over a field kk and A{\mathfrak A} a right HH-comodule algebra in the sense of Hausser and Nill. We first show that on the kk-vector space {\mathfrak A}\ot H^* we can define an algebra structure, denoted by {\mathfrak A}\ovsm H^*, in the monoidal category of left HH-modules (i.e. {\mathfrak A}\ovsm H^* is an HH-module algebra. Then we will prove that the category of two-sided (A,H)({\mathfrak A}, H)-bimodules \hba is isomorphic to the category of relative ({\mathfrak A}\ovsm H^*, H^*)-Hopf modules, as introduced in by Hausser and Nill. In the particular case where A=H{\mathfrak A}=H, we will obtain a result announced by Nill. We will also introduce the categories of Doi-Hopf modules and two-sided two-cosided Hopf modules and we will show that they are in certain situations isomorphic to module categories.Comment: 31 page

    Integrals for (dual) quasi-Hopf algebras. Applications

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    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\leq 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 the bijectivity of the antipode follows from the other axioms of a finite dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.Comment: 25 pages; new version with minor correction
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