A construction of semisimple tensor categories
type: Item
instance of: scholarly article
title: A construction of semisimple tensor categories
zbMATH Open document ID: 1097.18002
author: Friedrich Knop
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2006-08-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0605126
review text: Let \(K\) be a field of characteristic zero. The author extends \textit{P. Deligne}'s recent construction of a tensor category \({\mathcal R}ep(S_t,K)\) of representations of the ``symmetric group'' \(S_t\) with \(t\in K\) not necessarily a positive integer. Note that in case \(K\) is algebraically closed and \(t\in\mathbb N\), the tensor category \(\mathcal C_t\) of ordinary representations of \(S_t\) determines \(S_t\) up to isomorphism. In this case, \(\mathcal C_t\) is a specialization of \({\mathcal R}ep(S_t,K)\). For an abelian category \(\mathcal A\), the author constructs a semisimple tensor category \(\mathcal T(\mathcal A,K)\) in several steps. First, he extends the morphisms \(x\rightarrow y\) of \(\mathcal A\) to correspondences, i.e., equicalence classes of maps \(F\colon c\rightarrow x\oplus y\), where \(F\colon c\rightarrow x\oplus y\) and \(G\colon d\rightarrow x\oplus y\) are equivalent if and only if \(\text{Im}\, F=\text{Im}\, G\), and the kernels of \(F\) and \(G\) coincide in the Grothendieck monoid of \(\mathcal A\). The composition of correspondences is defined in the obvious way, like in Deligne's paper. Then the morphisms are extended to \(\mathbb Z\)-linear combinations of correspondences. The biproduct in the category \(\mathcal T_1(\mathcal A)\) thus obtained gives rise to a symmetric monoidal structure with unital object 0. Moreover, \(\mathcal T_1(\mathcal A)\) is \(\mathbb Z[\text{simp}(\mathcal A)]\)-linear, where \(\text{simp}(\mathcal A)\) is a representative system of the isomorphism classes of simple objects in \(\mathcal A\). To make the category \(K\)-linear, a homomorphism \(t\colon\mathbb Z[\text{simp}(\mathcal A)] \rightarrow K\) has to be fixed, such that \(K\) becomes a \(\mathbb Z[\text{simp}(\mathcal A)]\)-module. Finally, the category is made pseudo-abelian (SGA~4), i.e., additive with splitting idempotents, in a canonical way, which gives a category \(\mathcal T(\mathcal A,K)\) with a symmetric \(K\)-bilinear rigid tens…
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 5046014
MaRDI profile type: Publication
OpenAlex ID: W2006711657
arXiv ID: math/0605126
cites work: Diagram chasing in Mal'cev categories
cites work: Q5442216
cites work: Determinants on Semilattices
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cites work: Hadamard determinants Möbius functions, and the chromatic number of a graph
DOI: 10.1016/J.CRMA.2006.05.009
description: scientific article