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Serre theorem for involutory Hopf algebras. (English) Zbl 1201.16029

The Serre category which is a particular monoidal category is introduced first. Then the main result, the Serre theorem for involutory Hopf algebras is studied. Specifically, let \(H\) be an involutory Hopf algebra, \(M,N\) two \(H\)-(co)modules such that \(M\otimes N\) is (co)semisimple as an \(H\)-(co)module. If \(N\) (resp. \(M\)) is a finitely generated projective \(k\)-module with invertible Hattori-Stallings rank in the field \(k\) then \(M\) (resp. \(N\)) is (co)semisimple as an \(H\)-(co)module. In particular, the full subcategories of all finite dimensional modules, comodules or Yetter-Drinfel’d modules over \(H\) the dimensions of which are invertible in \(k\) are Serre categories.

MSC:

16T05 Hopf algebras and their applications
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
16T15 Coalgebras and comodules; corings
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References:

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