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Serre Theorem for involutory Hopf algebras

✍ Scribed by Gigel Militaru

Book ID
111488549
Publisher
SP Versita
Year
2010
Tongue
English
Weight
367 KB
Volume
8
Category
Article
ISSN
1895-1074

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✦ Synopsis

We call a monoidal category a Serre category if for any C , D ∈ such that C βŠ— D is semisimple, C and D are semisimple objects in . Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M βŠ— N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective -module with invertible Hattory-Stallings rank in then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel'd modules over H the dimension of which are invertible in are Serre categories.

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For a Hilbert C \* -module X over a C \* -algebra A, we introduce a vector bundle E X associated to X. We prove that E X has an hermitian metric and a flat connection. We introduce a vector space Ξ“ X of holomorphic sections of E X with the following properties: (i) Ξ“ X is a Hilbert A-module, (ii) th