Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andruskiewitsch N., Etingof P., Gelaki S., Triangular Hopf algebras with the Chevalley property, Michigan Math. J., 2001, 49, 277–298
Chevalley C., Theorie des Groupes de Lie, Vol. III, Hermann, Paris, 1954
Lambe L.A., Radford D., Algebraic aspects of the quantum Yang-Baxter equation, J. Algebra, 1992, 154, 222–288
Serre J.-P., Sur la semi-simplicitï’·e des produits tensoriels de reprï’·esentations de groupes, Invent. Math., 1994, 116, 513–530
Serre J.-P., Semisimplicity and tensor products of group representations: converse theorems, J. Algebra, 1997, 194, 496–520
Serre J.-P., Moursund Lectures 1998, Notes by Duckworth W.E., preprint availabe at http://math.uoregon.edu/resources/serre/
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Militaru, G. Serre Theorem for involutory Hopf algebras. centr.eur.j.math. 8, 15–21 (2010). https://doi.org/10.2478/s11533-009-0062-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-009-0062-z