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We extend Schaumann's theory of pivotal structures on fusion categoriesmatched to a module category and of module traces developed in arXiv:1206.5716to the case of non-semisimple tensor categories, and use it to studyeigenvalues of the squared antipode $S^2$ in weak Hopf algebras. In particular,we diagonalize $S^2$ for semisimple weak Hopf algebras in characteristic zero,generalizing the result of Nikshych in the pseudounitary case. We show that theanswer depends only on the Grothendieck group data of the pivotalizations ofthe categories involved and the global dimension of the fusion category (thus,all eigenvalues belong to the corresponding number field). On the other hand,we study the eigenvalues of $S^2$ on the non-semisimple weak Hopf algebrasattached to dynamical quantum groups at roots of $1$ defined by D. Nikshych andthe author in arXiv:math/0003221, and show that they depend nontrivially on thecontinuous parameters of the corresponding module category. We then computethese eigenvalues as rational functions of these parameters. The paper alsocontains an appendix by G. Schaumann discussing the connection between ourgeneralization of module traces and the notion of an inner product modulecategory introduced in arXiv:1405.5667.

Eigenvalues of the squared antipode in finite dimensional weak Hopf algebras