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Finitary Galois Extensions over noncommutative bases
, 2005
"... We study Galois extensions M (co)H ⊂ M for H(co)module algebras M if H is a Frobenius Hopf algebroid. The relation between the action and coaction pictures is analogous to that found in HopfGalois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various cla ..."
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We study Galois extensions M (co)H ⊂ M for H(co)module algebras M if H is a Frobenius Hopf algebroid. The relation between the action and coaction pictures is analogous to that found in HopfGalois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various classical theorems of KreimerTakeuchi, DoiTakeuchi and CohenFischmanMontgomery. We find that the Galois extensions N ⊂ M over some Frobenius Hopf algebroid are precisely the balanced depth 2 Frobenius extensions. We prove that the YetterDrinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by a slightly generalized BrzezińskiMilitaru Theorem. Contravariant ”fiber functors ” are used to prove an analogue of Ulbrich’s Theorem and to get a monoidal embedding of the module category ME of the endomorphism Hopf algebroid E = End N MN into N M op N.
SCALAR EXTENSION OF BICOALGEBROIDS
, 707
"... Abstract. After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided ju ..."
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Abstract. After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided just as in the case of bialgebroids, and is embedded into the one–sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of [2] and [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule