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Triangular Hopf algebras with the Chevalley property. (English) Zbl 1016.16029

The authors work over the field of complex numbers. A Hopf algebra \(H\) is said to have the Chevalley property if the tensor product of any two simple \(H\)-modules is semisimple. The main aim of the paper is to classify finite dimensional triangular Hopf algebras with the Chevalley property. More precisely, it is proved that a finite dimensional triangular Hopf algebra \(H\) has the Chevalley property if and only if it is a twist of a finite dimensional triangular Hopf algebra \(A\) with \(R\)-matrix of rank \(\leq 2\). Moreover, it is proved that such a Hopf algebra \(A\) is obtained from a cocommutative Hopf superalgebra (i.e. the group algebra of a supergroup) by a certain modification of the comultiplication. As an application of the main theorem, it is proved that a finite dimensional cotriangular pointed Hopf algebra is generated by grouplike elements and skewprimitive elements. Finally it is proved that in any linear Abelian symmetric rigid tensor category with finitely many irreducible objects (in particular in the category of representations over a finite dimensional triangular Hopf algebra), the categorical dimensions of objects are integers. The paper contains many examples and several interesting open questions.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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