A construction of semisimple tensor categories

Friedrich Knop

doi:10.1016/j.crma.2006.05.009

Algebra

A construction of semisimple tensor categories
[Une construction des catégories tensorielles semi-simples]

Présenté par : Pierre Deligne

Friedrich Knop ¹

¹ Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA

Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 15-18.

Résumé
Abstract

Soit $A$ une catégorie abélienne dont chaque objet n'a qu'un nombre fini de sous-objets. A partir de $A$ on construit une catégorie tensorielle semi-simple $T$ . On démontre que $T$ interpole les catégories $Rep (Aut (p), K)$ où p parcourt certains pro-objets projectifs de $A$ . Ceci étend une construction de Deligne pour les groupes symétriques.

Let $A$ be an Abelian category such that every object has only finitely many subobjects. From $A$ we construct a semisimple tensor category $T$ . We show that $T$ interpolates the categories $Rep (Aut (p), K)$ where p runs through certain projective pro-objects of $A$ . This extends a construction of Deligne for symmetric groups.

Reçu le : 2006-05-04
Accepté le : 2006-05-15
Publié le : 2006-06-13

DOI : 10.1016/j.crma.2006.05.009

Affiliations des auteurs :

Friedrich Knop ¹

¹ Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA

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Friedrich Knop. A construction of semisimple tensor categories. Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 15-18. doi : 10.1016/j.crma.2006.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.009/

Bibliographie
Cité par

[1] A. Carboni; J. Lambek; M. Pedicchio Diagram chasing in Mal'cev categories, J. Pure Appl. Algebra, Volume 69 (1991), pp. 271-284

[2] P. Deligne La catégorie des représentations du groupe symétrique $S_{t}$ lorsque t n'est pas un entier naturel http://math.ias.edu/~phares/deligne/Symetrique.pdf Preprint (78 p.)

[3] B. Lindström Determinants on semilattices, Proc. Amer. Math. Soc., Volume 20 (1969), pp. 207-208

[4] R. Stanley Modular elements of geometric lattices, Algebra Universalis, Volume 1 (1971/72), pp. 214-217

[5] R. Stanley Supersolvable lattices, Algebra Universalis, Volume 2 (1972), pp. 197-217

[6] H. Wilf Hadamard determinants, Möbius functions, and the chromatic number of a graph, Bull. Amer. Math. Soc., Volume 74 (1968), pp. 960-964

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