A modified trace for a finite \({\mathbb {k}}\)-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. The modified trace provides a meaningful generalisation of the categorical trace to non-semisimple categories and allows to construct interesting topological invariants. We show that a non-degenerate modified trace defines a compatible with duality Calabi–Yau structure on the subcategory of projective objects. We prove, that for any finite-dimensional unimodular pivotal Hopf algebra over a field \({\mathbb {k}}\), a modified trace is determined by a symmetric linear form on the Hopf algebra constructed from an integral. More precisely, we prove that shifting with the pivotal element defines an isomorphism between the space of right integrals, which is known to be 1-dimensional, and the space of modified traces. This result allows us to compute modified traces for all simply laced restricted quantum groups at roots of unity.

有限\（{{mathbb {k}} \）线性枢轴类别的修改轨迹是投影对象的内同态空间上的一族线性形式，具有周期性和所谓的部分轨迹特性。修改后的迹线将分类迹线有意义地概括为非半简单类别，并允许构造有趣的拓扑不变量。我们显示了一个非简并的修饰轨迹，在射影对象的子类别上定义了与对偶Calabi-Yau结构兼容。我们证明，对于域\（{{mathbb {k}} \）上的任何有限维单模枢轴Hopf代数，修改后的迹线由Hopf代数上由整数构造的对称线性形式确定。更确切地说，我们证明了通过枢轴元素移动会在已知为一维的右积分空间与修饰迹线的空间之间定义同构。这个结果使我们能够计算出所有统一根上的简单带限位量子群的修饰迹线。