We call a monoidal category C a Serre category if for any C, D ∈ C such that C ⊗ D is semisimple, C and D are semisimple objects in C. Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M ⊗ N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective k-module with invertible Hattory-Stallings rank in k then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel’d modules over H the dimension of which are invertible in k are Serre categories.
@article{bwmeta1.element.doi-10_2478_s11533-009-0062-z,
author = {Gigel Militaru},
title = {Serre Theorem for involutory Hopf algebras},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {15-21},
zbl = {1201.16029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0062-z}
}
Gigel Militaru. Serre Theorem for involutory Hopf algebras. Open Mathematics, Tome 8 (2010) pp. 15-21. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0062-z/
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