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

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

2002-11-27

arxiv.org/abs/math/0211430v1

Mathematics

Quantum Algebra

31 pages

Scientific paper

Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and ${\mathfrak A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space ${\mathfrak A}\ot H^*$ we can define an algebra structure, denoted by ${\mathfrak A}\ovsm H^*$, in the monoidal category of left $H$-modules (i.e. ${\mathfrak A}\ovsm H^*$ is an $H$-module algebra. Then we will prove that the category of two-sided $({\mathfrak A}, H)$-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 ${\mathfrak A}=H$, 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.

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