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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:
16W30Hopf algebras (associative rings and algebras) (MSC2000)
18D10Monoidal, symmetric monoidal and braided categories
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References:
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