The relative modular object and Frobenius extensions of finite Hopf algebras (Q342829)

In this paper, the author considers a tensor functor \(F:\mathcal C\rightarrow \mathcal D\) between two finite tensor categories \(\mathcal C\) and \(\mathcal D\) having a left adjoint \(L\) and a right adjoint \(R\). He shows that \(L\) has a left adjoint, if and only if \(R\) has a right adjoint, if and only if there is an object \(\chi_F\in \mathcal D\), called the relative modular object of \(F\), such that \(R\cong L(-\otimes\chi_F)\). The object \(\chi_F\) is unique up to isomorphisms, and the main result of the paper (Theorem \(4.8\)) gives a formula for the relative modular object. As an application of this result, after giving an explicit expression of the modular object in the category \(\mathcal{\nu}_H\) of right modules over a Hopf algebra \(H\) in a braided finite tensor category \(\mathcal{\nu}\), the author obtains (Theorem \(5.5\)) a braided version of a result by \textit{D. Fischman} et al. [Trans. Am. Math. Soc. 349, No. 12, 4857--4895 (1997; Zbl 0899.16020)].