Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras

Abstract

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors − ⊗ _k V and V ⊗ _k  − are known to be left adjoint to some kind of Hom-functors as endofunctors of _H 𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case.

In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors − ⊗V and − ⊗V* depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors − ⊗V* and Hom _k (V, −) explicitly. Furthermore, we consider the special case V = A being an H-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.

Key Words:

• Comonad
• Module algebra
• Monad
• Monoidal category
• Quasi-Hopf algebra

2010 Mathematics Subject Classification:

• 16T05
• 18A40
• 18D10

ACKNOWLEDGMENT

The author would like to thank Professor Robert Wisbauer for his continuous support and encouragement. He is also grateful to the referee for helpful advice leading to an improvement of the presentation.

Notes

Communicated by V. A. Artamonov.