Abstract Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k , and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified traces are non-degenerate, and we show that non-degenerate left/right modified traces can only exist for unimodular H. This generalises results of Beliakova, Blanchet, and Gainutdinov [1] from Hopf algebras to quasi-Hopf algebras. As an example we compute cointegrals and modified traces for the family of symplectic fermion quasi-Hopf algebras.

中文翻译：

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拟 Hopf 代数的修正迹

摘要 令H 为域k 上的有限维单模枢轴拟Hopf 代数，并令H-mod 为有限维H 模的枢轴张量范畴。我们给出了射影模张量理想 H-pmod 上的左（右）修正迹与 H 的左（右）协整之间的双射。非零左/右修正迹是非退化的，我们表明非退化左/右修改迹只能存在于单模 H。这将 Beliakova、Blanchet 和 Gainutdinov [1] 的结果从 Hopf 代数推广到准 Hopf 代数。作为一个例子，我们计算辛费米子拟 Hopf 代数族的协积分和修正迹。