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Projective objects and the modified trace in factorisable finite tensor categories

Part of: Lie algebras and Lie superalgebras Categories and geometry Homological algebra Categories with structure

Published online by Cambridge University Press:  26 March 2020

Azat M. Gainutdinov  and
Ingo Runkel
Azat M. Gainutdinov
Affiliation:
Institut Denis Poisson, CNRS, Université de Tours, Université d’Orléans, Parc de Grandmont, 37200Tours, France email azat.gainutdinov@lmpt.univ-tours.fr Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146Hamburg, Germany
Ingo Runkel
Affiliation:
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146Hamburg, Germany email ingo.runkel@uni-hamburg.de
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Abstract

For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:

  1. (1) ${\mathcal{C}}$ always contains a simple projective object;

  2. (2) if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;

  3. (3) the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;

  4. (4) the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.

Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

Keywords

braided and pivotal categoriesprojective objectsmodified tracesGrothendieck ringsvertex-operator algebrasVerlinde-like formula

MSC classification

Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 18G05: Projectives and injectives 18F30: Grothendieck groups 17B69: Vertex operators; vertex operator algebras and related structures
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 4 , April 2020 , pp. 770 - 821
Copyright
© The Authors 2020

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