The pivotal cover and Frobenius–Schur indicators

In this paper, we introduce the notion of the pivotal cover CpivCpiv of a left rigid monoidal category CC to develop a theoretical foundation for the theory of Frobenius–Schur (FS) indicators in “non-pivotal” settings. For an object V∈CpivV∈Cpiv, the (n,r)(n,r)-th FS indicator νn,r(V)νn,r(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.