Abstract
We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category \({\mathscr{M}}\), we introduce the notion of a Σ-twisted trace on the class \(\text {Proj}({\mathscr{M}})\) of projective objects of \({\mathscr{M}}\). In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on \(\text {Proj}({\mathscr{M}})\) and the set of natural transformations from Σ to the Nakayama functor of \({\mathscr{M}}\). Non-degeneracy and compatibility with the module structure (when \({\mathscr{M}}\) is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.
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Acknowledgements
We are grateful to the referees for careful reading of the manuscript. The first author (T.S.) is supported by JSPS KAKENHI Grant Number JP19K14517. The second author (K.S.) is supported by JSPS KAKENHI Grant Number JP20K03520.
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Shibata, T., Shimizu, K. Modified Traces and the Nakayama Functor. Algebr Represent Theor 26, 513–551 (2023). https://doi.org/10.1007/s10468-021-10102-5
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DOI: https://doi.org/10.1007/s10468-021-10102-5