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Fusion categories arising from semisimple Lie algebras. (English) Zbl 0827.17010
There are some interesting categories for a given finite Cartan datum, for example, the category $${\mathcal O}$$ of certain finitely generated modules of the corresponding semisimple Lie algebra $${\mathfrak g}$$, the category of rational modules of the corresponding semisimple, simply connected algebraic group $$G$$ over a field of positive characteristic, the category of locally finite modules of the associated quantum algebra $$U$$ specialized at an $$l$$-th root of unity. In this paper, the authors investigate a certain semisimple subcategory equipped with a ‘reduced’ tensor product for the three above mentioned situations respectively. Also, they determine the fusion rules for this tensor product in each case via known character formulas for the involved modules.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 20G05 Representation theory for linear algebraic groups 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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