Serre Theorem for involutory Hopf algebras
Open Mathematics (2010)
- Volume: 8, Issue: 1, page 15-21
- ISSN: 2391-5455
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topGigel Militaru. "Serre Theorem for involutory Hopf algebras." Open Mathematics 8.1 (2010): 15-21. <http://eudml.org/doc/269159>.
@article{GigelMilitaru2010,
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.},
author = {Gigel Militaru},
journal = {Open Mathematics},
keywords = {Hopf algebras; Semisimple modules; Chevalley property; involutory Hopf algebras; semisimple modules; monoidal categories; Serre categories; comodules; finite dimensional modules; Yetter-Drinfeld modules},
language = {eng},
number = {1},
pages = {15-21},
title = {Serre Theorem for involutory Hopf algebras},
url = {http://eudml.org/doc/269159},
volume = {8},
year = {2010},
}
TY - JOUR
AU - Gigel Militaru
TI - Serre Theorem for involutory Hopf algebras
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 15
EP - 21
AB - 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.
LA - eng
KW - Hopf algebras; Semisimple modules; Chevalley property; involutory Hopf algebras; semisimple modules; monoidal categories; Serre categories; comodules; finite dimensional modules; Yetter-Drinfeld modules
UR - http://eudml.org/doc/269159
ER -
References
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- [2] Chevalley C., Theorie des Groupes de Lie, Vol. III, Hermann, Paris, 1954
- [3] Lambe L.A., Radford D., Algebraic aspects of the quantum Yang-Baxter equation, J. Algebra, 1992, 154, 222–288
- [4] Serre J.-P., Sur la semi-simplicitï’·e des produits tensoriels de reprï’·esentations de groupes, Invent. Math., 1994, 116, 513–530 http://dx.doi.org/10.1007/BF01231571
- [5] Serre J.-P., Semisimplicity and tensor products of group representations: converse theorems, J. Algebra, 1997, 194, 496–520 http://dx.doi.org/10.1006/jabr.1996.6929
- [6] Serre J.-P., Moursund Lectures 1998, Notes by Duckworth W.E., preprint availabe at http://math.uoregon.edu/resources/serre/
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