Fusion categories are fundamental objects in quantum algebra, but their definition is narrow in some respects. By definition a fusion category must be k-linear for some field k, and every simple object V is strongly simple, meaning that (V) = k. We prove that linearity follows automatically from semisimplicity: Every connected, finite, semisimple, rigid, monoidal category \C is k-linear and finite-dimensional for some field k. Barring inseparable extensions, such a category becomes a multifusion category after passing to an algebraic extension of k.
The proof depends on a result in Galois theory of independent interest, namely a finiteness theorem for abstract composita.
We give a 3-universal property for the Karoubi envelope of a 2-category. Using this, we show that the 3-categories of finite semisimple 2-categories (as introduced in arXiv:1812.11933) and of multifusion categories are equivalent.
Free products of semisimple tesnor categories are constructed with the help of polygonal presentation. The semisimplicity criterion is obtained for the Bisch-Jones' planar algebras as a byproduct.
Tannaka duals of Hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories, which are shown to include the Tannaka dual of the dual Hopf algebras. The second orbifolds are then canonically isomorphic to the initial tensor categories.
We give a classification of semisimple and separable algebras in a multi-fusion category over an arbitrary field in analogy to Wedderben-Artin theorem in classical algebras. It turns out that, if the multi-fusion category admits a semisimple Drinfeld center, the only obstruction to the separability of a semisimple algebra arises from inseparable field extensions as in classical algebras. Among others, we show that a division algebra is separable if and only if it has a nonvanishing dimension.
We classify semisimple module categories over the tensor category of representations of quantum SL(2) extending previous results to the roots of unity and positive characteristic cases.
Given an action of a finite group G on a fusion category C we give a criterion for the category of G-equivariant objects in C to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in math/0203060. Namely, we show that certain Z/2Z-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami are equivalent to representation categories of non gro…
We show that every essentially small finitely semisimple k-linear additive spherical category in which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as a Weak Hopf Algebra.
We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of…
One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to certain classes of Hopf algebras, e.g. to semisimple ones. Semisimple Hopf algebras deserve to be considered as "quantum" analogue of finite groups, but even so, the problem remains extremely hard (even in low dimensions) and very little is known. A great boost to the theory of Hopf algebras was given by Drinfeld who invented the fundamental class of (quasi)trian…
We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category C if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V|^2 of any simple object V of C. We show, using the Ito-Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categor…
We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the s…
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realize…
Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be block-diagonalized. A conjectural connection between the block-diagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models.
We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P.Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite.
Let λbe a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char(k)>n, and let V be an object of C. In our main result (Theorem 4.3) we introduce a finite set of integers F(λ) and prove that if the Schur functor \mathbb{S}_λ V of V is semisimple and the dimension of V is not in F(λ), then V is semisimple. Moreover, we prove that for each d in F(λ) there exist a symmetric rigid tensor category C over k and a non-semisimple object V in C of dimension d such that \mathbb{S}_λ V is semisimple (which shows that our result is the best…
Let $k$ be a field, and $H$ a Hopf algebra with bijective antipode. If $H$ is commutative, noetherian, semisimple and cosemisimple, then the category ${}_{H}{\mathcal {YD}}^H$ of Yetter-Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that $H$ is commutative.
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 eleme…
We prove some results on the structure of certain classes of integral fusion categories and semisimple Hopf algebras under restrictions on the set of its irreducible degrees.
We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same category.
A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.
This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions which preserve the ascending filtration on tensor algebras, and the second for actions which preserve the descending filtration on …
We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.
We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.
We use a version of Haboush's theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover, any isomorphism between such structures can be reduced modulo $p$. This fills a gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer-Pic…
In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists are roots of unity, and square of the braiding has finite order. We also discuss the notion of the qua…
We propose a conjectural extension to positive characteristic case of a well known Deligne's theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many isomorphism classes of simple objects.
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their associativity and commutativity morphisms.
Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, th…
We introduce a tensor category O_+ (resp. O_{-}) of certain modules of gl_{\infty} with non-negative (resp. non-positive) integral central charges with the usual tensor product. We also introduce a tensor category O_f consisting of certain modules over GL(N) for all N. We show that the tensor categories O_+, O_{-} and O_f are semisimple abelian and all equivalent to each other. We give a formula to decompose a tensor product of two modules in each of these categories. We also introduce a tensor category O^w of certain modules over W_{1 +\infty} with non-negative integral central charges. We sh…
We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field.
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely-generated and projective $\Bbbk$-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich's reconstruction theorem for Hopf algebras. When $\Bbbk$ is field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.
Given a braided pivotal category $\mathcal C$ and a pivotal module tensor category $\mathcal M$, we define a functor $\mathrm{Tr}_{\mathcal C}:\mathcal M \to \mathcal C$, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor $\mathrm{Tr}_{\mathcal C}$ comes equipped with natural isomorphisms $τ_{x,y}:\mathrm{Tr}_{\mathcal C}(x \otimes y) \to \mathrm{Tr}_{\mathcal C}(y \otimes x)$, which we call the traciators. This situation lends itself to a diagramatic calculus of `strings on cylinders', where the traciator corresponds to wrapping a string…
We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry algebras with additional structure, which in suitable cases is the one of a finite tensor category. The problem of specifying the correlators can then be encoded in algebraic structure internal to those categories. After reviewing results for conformal field theories for which these representation categories are semisimple, we explain what is known about represent…
After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and co-semisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction - restriction. For reasons of size, the sl(3) tables of induction are only given for theories with self-fusion (existence of a monoidal structure).
We use factorizable finite tensor categories, and specifically the representation categories of factorizable ribbon Hopf algebras H, as a laboratory for exploring bulk correlation functions in local logarithmic conformal field theories. For any ribbon Hopf algebra automorphism omega of H we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric Frobenius algebra. We derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters; as a crucial ingredient this involves the Cartan mat…
In this paper we provide a complete classification of fusion categories of Frobenius-Perron (FP) dimension pq, where p<q are distinct primes, thus giving a categorical generalization of math.QA/9801129. As a corollary we also obtain the classification of semisimple quasi-Hopf algebras of dimension pq. A concise formulation of our main result is: Let C be a fusion category over the complex numbers of FP dimension pq. Then either p=2 and C is a Tambara-Yamagami category of dimension 2q, or C is group-theoretical in the sense of math.QA/0203060 (which easily yields the full classification). As a …
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability under extension of the base field. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference of the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose r…
