The bicategories of corings

To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projective on one side, together with a map of corings over the same base ring, lead to the notion of a module-morphism, which extends the notion of a morphism of corings (over different base rings). A module-morphism of corings induces functors between the categories of comodules. These functors are termed pull-back and push-out functors, respectively, and thus relate categories of comodules of different corings. We study when the pull-back functor is fully faithful and when it is an equivalence. A generalised descent associated to a morphism of corings is introduced. We define a category of module-morphisms, and show that push-out functors are naturally isomorphic to each other if and only if the corresponding module-morphisms are mutually isomorphic. All these topics are studied within a unifying language of bicategories and the extensive use is made of interpretation of corings as comonads in the bicategory Bim of bimodules and module-morphisms as 1-cells in the associated bicategories of comonads in Bim.     

A note on coring extensions

A notion of a coring extension is defined and it is shown to be equivalent to the existence of an additive functor between comodule categories that factorises through forgetful functors. This correspondence between coring extensions and factorisable functors is illustrated by functors between categories of descent data. A category in which objects are corings and morphisms are coring extensions is also introduced.

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Sunto Si fornisce una definizione per estensioni di coanelli e si dimostra l'equivalenza di tale definizione con l'esistenza di un funtore additivo tra categorie di comoduli che si fattorizzi attraverso il funtore dimenticante. Questa corrispondenza tra estensioni di coanelli e funtori fattorizzabili è illustrata da funtori tra categorie di discesa. Si introduce inoltre una categoria i cui oggetti sono coanelli e i morfismi sono estensioni di coanelli.

     

Co-Frobenius corings and adjoint functors

We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Morita contexts to study Frobenius properties of corings and a second kind of Morita contexts to study adjoint pairs. Comparing both Morita contexts, we obtain our main result that characterizes quasi-co-Frobenius corings in terms of a pair of adjoint functors (F,G) such that (G,F) is locally quasi-adjoint in a sense defined in this note.     

Equivalences of comodule categories for coalgebras over rings

In this article we defined and studied quasi-finite comodules, the cohom functors for coalgebras over rings. Linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work characterizes equivalences between comodule categories generalizing the Morita-Takeuchi theory to coalgebras over rings. Morita-Takeuchi contexts in our setting is defined and investigated, a correspondence between strict Morita-Takeuchi contexts and equivalences of comodule categories over the involved coalgebras is obtained. Finally, we proved that for coalgebras over QF-rings Takeuchi's representation of the cohom functor is also valid.     

Bialgebroid actions on depth two extensions and duality

A general notion of depth two for ring homomorphism N→M is introduced. The step two centralizers A=End_NM_N and in the Jones tower above N→M are shown in a natural way via H-equivalence to be dual bimodules for Morita equivalent endomorphism rings, the step one and three centralizers, R=C_M(N) and C=End_N-M(M⊗_NM). We show A and B to possess dual left and right R-bialgebroid structures which generalize Lu's fundamental bialgebroids over an algebra. There are actions of A and B on M and with Galois properties. If M|N is depth two and Frobenius with R a separable algebra, we show that A and B are dual weak Hopf algebras fitting into a duality-for-actions tower extending previous results in this area for subfactors and Frobenius extensions.     

Finitely semisimple spherical categories and modular categories are self-dual

We show that every essentially small finitely semisimple k-linear additive spherical category for 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.     

Localization of Rota–Baxter algebras

A commutative Rota–Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota–Baxter algebras, we extend the central concept of localization for commutative algebras to commutative Rota–Baxter algebras. The existence of such a localization is proved and, under mild conditions, its explicit construction is obtained. The existence of tensor products of commutative Rota–Baxter algebras is also proved and the compatibility of localization and the tensor product of Rota–Baxter algebras is established. We further study Rota–Baxter coverings and show that they form a Grothendieck topology.     

Full ideals of polynomial rings

An idealI of the ringK[x ₁, ...,x _n ] of polynomials over a fieldK inn indeterminates is a full ideal ifI is closed under substitution,f I,g ₁...g_n K[x ₁, ...,x _n ] implyf(g ₁, ...,g _n ) I. In this paper we continue the investigation of full ideals ofK[x ₁, ...,x _n ]. In particular we determine several classes of full ideals ofK[x, y] (K a finite field) and investigate properties of these classes.The first author gratefully acknowledges support from theDeutsche Forschungsgemeinschaft     

Making non-trivially associated tensor categories from left coset representatives

The paper begins by giving an algebraic structure on a set of coset representatives for the left action of a subgroup on a group. From this we construct a non-trivially associated tensor category. Also a double construction is given, and this allows the construction of a non-trivially associated braided tensor category. In this category we explicitly reconstruct a braided Hopf algebra, whose representations comprise the category itself.     

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C_c^∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c^∞(G)-C_c^∞(H)-bimodule C_c^∞(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c^∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.     

On invariants of modular categories beyond modular data

We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the W-matrix—the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the W-matrix and the set of punctured S-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.     

Invertible bimodule categories over the representation category of a Hopf algebra

For any finite-dimensional Hopf algebra H   we construct a group homomorphism BiGal(H)→BrPic(Rep(H))$\mathrm{BiGal}(H)\to \text{BrPic}(\mathrm{Rep}(H))$, from the group of equivalence classes of H  -biGalois objects to the group of equivalence classes of invertible exact Rep(H)$\mathrm{Rep}(H)$-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H=_Tq$H=T_q$ is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(_Tq)$\mathrm{Rep}(T_q)$-bimodule categories.     

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.     

A remark on sums of units

For everyn≧2 we construct a factorial domainR for whichn minimal with the property that every element can be written as the sum of at mostn units.     

Plethystic algebra

The notion of a Z-algebra has a non-linear analogue, whose purpose it is to control operations on commutative rings rather than linear operations on abelian groups. These plethories can also be considered non-linear generalizations of cocommutative bialgebras. We establish a number of category-theoretic facts about plethories and their actions, including a Tannaka-Krein-style reconstruction theorem. We show that the classical ring of Witt vectors, with all its concomitant structure, can be understood in a formula-free way in terms of a plethystic version of an affine blow-up applied to the plethory generated by the Frobenius map. We also discuss the linear and infinitesimal structure of plethories and explain how this gives Bloch's Frobenius operator on the de Rham-Witt complex.     

The representation-theoretic rank of the doubles of quasi-quantum groups

We compute the representation-theoretic rank of a finite dimensional quasi-Hopf algebra H and of its quantum double D(H), within the rigid braided category of finite dimensional left D(H)-modules.     

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with gr(A) in RG(p), where gr(A) is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_±(p), A(q), or H(32), constructed in [5] and [8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.     

On radicals of module coalgebras

We introduce the notion of an idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming a bijective antipode.     

On the central charge of a factorizable Hopf algebra

For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel’d element differs from the value that it takes on the Drinfel’d element itself by at most a fourth root of unity. This can be reformulated by saying that the central charge of the Hopf algebra is an integer. If the dimension of the Hopf algebra is odd, we show that these two values differ by at most a sign, which can be reformulated by saying that the central charge is even. We give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs. To formulate our results, we use the language of modular data.     

The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras

With an aim of exploring homological algebra for weak Hopf modules, this paper investigates the HOM-functor and presents the structure theorem for endomorphism algebras of weak two-sided (A,H)-Hopf modules, and gives the duality theorem for weak “big” smash products.