Determinant and rank functions in semisimple pivotal Ab-categories

Document Type : Research Paper

Authors

Mathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.

Abstract

We investigate and generalize quantum determinants to semisimple spherical and pivotal categories. It is well known that traces are preserved by strong tensor functors; we show on one hand that in fact, weaker conditions on a functor are sufficient to continue preserving traces. On the other hand, we prove that these determinants are well-behaved under strong tensor functors. Further, we introduce a notion of domination rank for objects of a semisimple pivotal category and prove similar properties of the ordinary case. Furthermore, we expand the determinantal and McCoy ranks to introduce a morphism quantum rank function on a semisimple pivotal category.

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References

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Categories and General Algebraic Structures with Applications
Volume 19, Issue 1
August 2023
Pages 47-80

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