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Fusion categories for Chevalley groups. (Catégorie de fusion pour les groupes de Chevalley.) (French) Zbl 0801.20014
Let \(J\) be a Dynkin diagram and \(\theta\) an automorphism of \(J\). Let \(\mathfrak g\) be the simple Lie algebra over \(\mathbb{C}\) with Dynkin diagram \(J\) and \(\theta\) be the automorphism of \(\mathfrak g\) corresponding to \(\theta\). Let \(I = J/\theta\) denote the Dynkin diagram of \({\mathfrak g}^ \theta\) and let \(g\) be the number of Coxeter duals of \({\mathfrak g}^ \theta\).
In this paper, for each Chevalley group of the type associated with \(I\) over a field of characteristic \(p\), a rigid tensor category \({\mathcal P}_{\text{mod}}\) is defined. Moreover, it is shown that the Grothendieck ring of \({\mathcal P}_{\text{mod}}\) is isomorphic to the Grothendieck ring of the tensor category of all \(\mathcal O\)-integrable representations of the twisted loop algebra associated with \(J\) and \(\theta\) at level \(p-g\) and the truth of the conjecture linking quantum group representations and the representations of the twisted loop algebras is explained.

MSC:
20G05 Representation theory for linear algebraic groups
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20C33 Representations of finite groups of Lie type
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