Abstract
We discuss quasi-Hopf algebras as introduced by Drinfeld and generalize the Hom-tensor adjunctions from the Hopf case to the quasi-Hopf setting, making the module category over a quasi-Hopf algebra H into a biclosed monoidal category. However, in this case, the unit and counit of the adjunction are not trivial and should be suitably modified in terms of the reassociator and the quasi-antipode of the quasi-Hopf algebra H.
In a more general case, for a comodule algebra \( \mathcal{B} \) over a quasi-Hopf algebra H, the module category over \( \mathcal{B} \) need not to be monoidal. However, there is an action of a monoidal category on it. Using this action, we consider some kind of tensor and Hom-endofunctors of module category over \( \mathcal{B} \) and generalize some Hom-tensor relations from module category on H to this module category.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 74, Proceedings of the International Conference “Modern Algebra and Its Applications” (Batumi, 2010), Part 1, 2011.
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Bagheri, S. Hom-tensor relations for (quasi-) comodule algebras. J Math Sci 186, 701–705 (2012). https://doi.org/10.1007/s10958-012-1017-7
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DOI: https://doi.org/10.1007/s10958-012-1017-7