Abstract
We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra H, we give an explicit description of categorical cointegrals of the category MH of left H-modules in terms of cointegrals on H. Provided that H is unimodular, we also express the Frobenius structure of the ‘adjoint algebra’ in the Yetter-Drinfeld category YHHD by using an integral in H and a cointegral on H. Finally, we give a description of the twisted module trace for projective H-modules in terms of cointegrals on H.
Highlights
YD by using an integral in H and a cointegral on H
Results on Hopf algebras have been reexamined from the viewpoint of the theory of tensor categories since a result on Hopf algebras generalized to tensor categories is expected to be useful in applications to, e.g., low-dimensional topology and conformal field theories
Integrals and cointegrals for Hopf algebras are introduced by Sweedler [Swe69] and play an important role in the study of Hopf algebras
Summary
Results on Hopf algebras have been reexamined from the viewpoint of the theory of tensor categories since a result on Hopf algebras generalized to tensor categories is expected to be useful in applications to, e.g., low-dimensional topology and conformal field theories. As shown in [IM14], the Frobenius structure of A is written in terms of integrals and cointegrals of H if C = H Mfd is the category of finite-dimensional left H-modules. We discuss these problems in the case where C is the category H M of left modules over a finite-dimensional quasi-Hopf algebra H over a field k.
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