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The pivotal cover and Frobenius-Schur indicators

In this paper, we introduce the notion of the pivotal cover $\mathcal{C}^{\mathsf{piv}}$ of a left rigid monoidal category $\mathcal{C}$ to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object $\mathbf{V} \in \mathcal{C}^{\mathsf{piv}}$, the $(n, r)$-th FS indicator $ν_{n, r}(\mathbf{V})$ is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators. Based on our framework, we also study the FS indicators of the "adjoint object" in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.