We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups u q (sl 2 ).
We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category B with an action of a group G, we construct a braided G-crossed monoidal category ZG(B) with trivial component the Drinfeld center of B. We prove that, in the case of a G-action on the 2-category of representation of a tensor category C, the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of C. Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in [8].
For any finite-dimensional Hopf algebra H we construct a group homomorphism BiGal (H) → BrPic(Rep(H)), from the group of equivalence classes of H-biGalois objects to the group of equivalence classes of invertible exact Rep(H)-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H = T q is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(T q )bimodule categories. Subject Classification (2010): 18D10, 16W30, 19D23.
Mathematics
We classify exact indecomposable module categories over the representation category of all nontrivial Hopf algebras with coradical ޓ 3 and ޓ 4. As a byproduct, we compute all its Hopf-Galois extensions and we show that these Hopf algebras are cocycle deformations of their graded versions.
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