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Semisimple weak Hopf algebras. (English) Zbl 1066.16042

The paper initiates a systematic study of semisimple weak Hopf algebras. Analogues of several results for semisimple Hopf algebras are obtained. A canonical left integral on a weak Hopf algebra \(A\) is defined and used to characterize the semisimplicity of \(A\). Let \(A\) be a semisimple weak Hopf algebra. The quantum and Frobenius-Perron dimensions are studied for finite dimensional \(A\)-modules. The Grothendieck ring of \(A\) is studied and the class equation is extended for \(A\). Analogues of the trace formulas of Larson and Radford are proved and used to show that the categorical dimension of \(A\) divides its Frobenius-Perron dimension in the ring of algebraic integers. Module algebras and coalgebras over \(A\) are studied. The author indicates that this paper is a step in the direction of classifying semisimple weak Hopf algebras. This classification should include the classification of fusion categories and module categories over them.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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