Module categories, weak Hopf algebras and modular invariants

By V. Ostrik. 2001

644 citations recorded by the source corpus.

AbstractWe develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and we classify module categories over the fusion category of sl(2) at a positive integer level where we meet once again the ADE classification pattern.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d4a6cc616054f06f30494774890fe52522ad915

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On $G$--equivariant modular categories (2004, background): “e deﬁnition that C/Ghas a natural structure of a module category over the category RepGof ﬁnite-dimensional complex G-modules: for a representation ρ: G→EndV, we deﬁne ρ⊗(X,{ϕ}) = (V ⊗X,{ρg⊗ϕg}) (see [O] for overview of the notion of module category). The notation C/Gis justiﬁed by the following lem”
On the Structure of Modular Categories (2002, result): “d by quantum groupoids and by semisimple spherical categories, for which modularity of the quantum double was proven 2 in [31] and [29], respectively. (Actually, by results of Hayashi [17] and Ostrik [35], the categories considered in [29] are always representation categories of quantum groupoids. T”
On the Structure of Weak Hopf Algebras (2001, neutral): “Q A ] 1 J un 2”
WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES (2008, methodology): “For basics on these topics, we refer the reader to [BK, O1 , ENO].4 However, for reader’s convenience, we recall some of the most important definitions and facts that are used below.”
Nilpotent fusion categories (2006, background): “re ﬁnite, abelian, semisimple, and k-linear. All rings and modules have ﬁnite Z-rank. 2.1. Based rings and modules. Let Z+ be the semi-ringof non-negativeintegers. We will recall some deﬁnitions from [O]. Let R be a ring with identity which is a ﬁnite rank Z-module. A Z+-basis of R is a basis B = {X”
Fusion subcategories of representation categories of twisted quantum doubles of finite groups (2008, background): “(See [ O1 ] and [M3] for notions of module category and duality.)”
TFT construction of RCFT correlators: III: simple currents (2004, background): “Hom(1,A) = . (3.1) A is called simple i [ 24 ,40]”
Categorifications and cyclotomic rational double affine Hecke algebras (2013, background): “This means that O K is a left and right module category over O +,κ K , see [38] for”
On finite symmetries and their gauging in two dimensions (2017, background): “Let us describe them in more detail in the case of SU(2) WZW models, following [28].”