Finite tensor categories

By Pavel Etingof, V. Ostrik. 2003

85 citations recorded by the source corpus.

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive elements in a finite dimensional Hopf algebra in zero characteristic. We also develop the theory of module categories and dual categories for not necessarily semisimple finite tensor categories; the crucial new notion here is that of an exact module category. Finally, we classify indecomposable exact module categories over the simplest finite tensor categories, such as representations of a finite group in positive characteristic, representations of a finite supergroup, and representations of the Taft Hopf algebra.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e5503dceab6b094f9c22ea61e4cfc48b041c25b6

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Citations in the corpus

On the classification of finite-dimensional pointed Hopf algebras (2005, background): “s not understood. We now give a brief overview of the classiﬁcation program for ﬁnitedimensional Hopf algebras. We remark that these Hopf algebras give rise to ﬁnite tensor categories in the sense of [EO] and thus classiﬁcation results on ﬁnite-dimensional Hopf algebras should have applications in c”
Monoidal 2-structure of bimodule categories (2009, unknown): “hat ∆Γ = idand that Γ∆ is naturally equivalent to idvia fℓ. Finally, it is easy to see that Γ is a strict Z(C)-bimodule functor. As a corollary we get a well known result which appears for example in [EO04]. Corollary 7.11. (C ⊠Cop)∗ C ≃ Z(C) canonically as monoidal categories. 7.2. The 2-categories”