weak c cleft extensions weak entwining structures and weak hopf algebras abstract we formulate the concept of weak cleft extension for a weak entwining structure in a braided monoidal category c with equalizers and coequalizers we prove that if a is a weak c cleft extension then there is an isomorphism of algebras between a and a subobject of the tensor product of a c and c where a c is a subalgebra of a also we prove the corresponding dual results and linking the information of this two parts we obtain a general property for a pair morphisms f c a and g a c of algebras and coalgebras satisfying certain conditions finally as particular instances we get the results of fernandez and rodriguez the theorems of radford majid and bespalov in the case of hopf algebras with projection and the ones obtained by alonso and gonzalez for weak hopf algebras living in a symmetric category with split idempotents for example the weak theorem of blattner cohen and montgomery for weak hopf algebras with coalgebra splitting is one of them

a higher frobenius schur indicator formula for group theoretical fusion categories group theoretical fusion categories are defined by data concerning finite groups and their cohomology a finite group g endowed with a three cocycle and a subgroup h subset g endowed with a two cochain whose coboundary is the restriction of the objects of the category are g graded vector spaces with suitably twisted h actions the associativity of tensor products is controlled by simple objects are parametrized in terms of projective representations of finite groups namely of the stabilizers in h of right h cosets in g with respect to two cocycles defined by the initial data we derive and study general formulas that express the higher frobenius schur indicators of simple objects in a group theoretical fusion category in terms of the group theoretical and cohomological data defining the category and describing its simples

test of three dimensional q2r ising algorithm simulation results of q2r cellular automata on the simple cubic lattice are compared with usual monte carlo simulations of the ising model curie temperature spontaneous magnetisation and specific heat agree but at low temperatures the susceptibility and the distribution of local energies differ

drinfeld center of enriched monoidal categories abstract we define the drinfeld center of a monoidal category enriched over a braided monoidal category and show that every modular tensor category can be realized in a canonical way as the drinfeld center of a self enriched monoidal category we also give a generalization of this result for important applications in physics

on the frobenius schur indicators for quasi hopf algebras abstract mason and ng have given a generalization to semisimple quasi hopf algebras of linchenko and montgomery s generalization to semisimple hopf algebras of the classical frobenius schur theorem for group representations we give a simplified proof in particular a somewhat conceptual derivation of the appropriate form of the frobenius schur indicator that indicates if and in which of two possible fashions a given simple module is self dual

corrected confidence intervals after sequential testing with applications to survival analysis corrected confidence intervals are developed for an unknown parameter for data from a sequential experiment the corrections are obtained from asymptotic expansions for the distribution of the signed root transformation and they have a simple form the model does not require independent and identically distributed random variables or smooth stopping boundaries inference about the failure rate of an exponential distribution when data are subject to random censoring censoring by real time and optional stopping is used to illustrate the main results the accuracy of the method is confirmed through simulation studies

two sided two cosided hopf modules and doi hopf modules for quasi hopf algebras let h be a finite dimensional quasi hopf algebra over a field k and a a right h comodule algebra we introduce the category of two sided hopf modules and prove that it is isomorphic to a module category we also show that two sided hopf modules are coalgebra over a certain comonad we introduce doi hopf modules and show that they are comodules over a certain coring if the underlying h module coalgebra is finite dimensional then doi hopf modules are modules over a certain smash products a similar result holds for two sided two cosided hopf modules

radford s biproducts and yetter drinfeld modules for monoidal hom hopf algebras let h be a monoidal hom bialgebra and b be a left h hom module algebra and also a left h hom comodule coalgebra then in this paper we first introduce the notion of a hom smash coproduct which is a monoidal hom coalgebra second we find sufficient and necessary conditions for the hom smash product algebra structure and the hom smash coproduct coalgebra structure on b h to afford b h a monoidal hom bialgebra structure generalizing the well known radford s biproduct where the conditions are equivalent to that b is a bialgebra in the category of hom yetter drinfeld modules hydhh finally we illustrate the category of hom yetter drinfeld modules hydhh and prove that the category hydhh is a braided monoidal category

two characterizations of finite quasi hopf algebras abstract let h be a finite dimensional quasibialgebra we show that h is a quasi hopf algebra if and only if the monoidal category of its finite dimensional left modules is rigid if and only if a structure theorem for hopf modules over h holds

two applications of analytic functors we apply the theory of analytic functors to two topics related to theoretical computer science one is a mathematical foundation of certain syntactic well quasi orders and well orders appearing in graph theory the theory of term rewriting systems and proof theory the other is a new verification of the lagrange good inversion formula using several ideas appearing in semantics of lambda calculi especially the relation between categorical traces and fixpoint operators copyright 2002 elsevier science b v

exact sequences hochschild cohomology and the lie module structure over the m relative center abstract in this article we present actions by central elements on hochschild cohomology groups with arbitrary bimodule coefficients as well as an interpretation of these actions in terms of exact sequences since our construction utilizes the monoidal structure that the category of bimodules possesses we will further recognize that these actions are compatible with monoidal functors and thus as a consequence are invariant under morita equivalences by specializing the bimodule coefficients to the underlying algebra itself our efforts in particular yield a description of the degree n 0 part of the lie bracket in hochschild cohomology and thereby close a gap in earlier work by s schwede

milnor moore categories and monadic decomposition abstract in this paper monoidal hom lie algebras lie color algebras lie superalgebras and other type of generalized lie algebras are recovered by means of an iterated construction known as monadic decomposition of functors which is based on eilenberg moore categories to this aim we introduce the notion of milnor moore category as a monoidal category for which a milnor moore type theorem holds we also show how to lift the property of being a milnor moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

hopf bimodules coquasibialgebras and an exact sequence of kac abstract based on the ideas of tannaka krein reconstruction we present a categorical construction that assigns to any cleft hopf algebra inclusion k h a coquasibialgebra having k as a hopf subalgebra as a special case the construction gives an intrinsic connection between the bismash product k q and the double cross product q k constructed from the same combinatorial data a cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a sweedler three cocycle thus our construction assigns to every bicrossproduct or hopf algebra extension of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a sweedler three cocycle based on this observation we use the construction to prove generalizations of kac s exact sequence for the group of hopf algebra extensions of a group algebra by a dual group algebra

asymptotic expansions for the distributions of stopped random walks and first passage times let s n x 1 cdots x n n geq 1 be a d dimensional random walk and let t a inf n geq n a ng s n n geq a where n a o a let theta g ex 1 hat theta n g s n n and delta a t a hat theta t a a edgeworth type expansions are developed for p t a n y 1 leq delta a leq y 2 and for the distribution functions of t a and of sqrt t a h hat theta t a h theta where h is a real valued function such that h theta neq 0

idempotents and one sided units lattice invariants and a semigroup of functors on the category of monoids abstract for a monoid m we denote by g m the group of units e m the submonoid generated by the idempotents and g l m and g r m the submonoids consisting of all left or right units writing m for the monoidal category of monoids g e g l and g r are all monoidal functors m m there are other natural functors associated to submonoids generated by combinations of idempotents and one or two sided units the above functors generate a monoid with composition as its operation we show that this monoid has size 15 and describe its algebraic structure we also show how to associate certain lattice invariants to a monoid and classify the lattices that arise in this fashion a number of examples are discussed throughout some of which are essential for the proofs of the main theoretical results

symmetries and the u condition in hom yetter drinfeld categories let h s be a monoidal hom hopf algebra and hydhh the hom yetter drinfeld category over h then in this paper we first find sufficient and necessary conditions for hydhh to be symmetric and pseudosymmetric respectively second we study the u condition in hydhh and show that the hom yetter drinfeld module h adjoint resp h m coadjoint satisfies the u condition if and only if s2 id finally we prove that hydhh over a triangular resp cotriangular hom hopf algebra contains a rich symmetric subcategory

butterflies ii torsors for 2 group stacks abstract we study torsors over 2 groups and their morphisms in particular we study the first non abelian cohomology group with values in a 2 group butterfly diagrams encode morphisms of 2 groups and we employ them to examine the functorial behavior of non abelian cohomology under change of coefficients we re interpret the first non abelian cohomology with coefficients in a 2 group in terms of gerbes bound by a crossed module our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the fraction weak morphism determined by a butterfly as a practical byproduct we show how butterflies can be used to obtain explicit maps at the cocycle level in addition we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2 groups as well as specific features pertaining to group extensions

module categories of finite hopf algebroids and self duality we characterize the module categories of suitably finite hopf algebroids more precisely x r bialgebras in the sense of takeuchi 1977 that are hopf and finite in the sense of a work by the author 2000 as those k linear abelian monoidal categories that are module categories of some algebra and admit dual objects for sufficiently many of their objects then we proceed to show that in many situations the hopf algebroid can be chosen to be self dual in a sense to be made precise this generalizes a result of pfeiffer for pivotal fusion categories and the weak hopf algebras associated to them

modified traces for quasi hopf algebras abstract let h be a finite dimensional unimodular pivotal quasi hopf algebra over a field k and let h mod be the pivotal tensor category of finite dimensional h modules we give a bijection between left resp right modified traces on the tensor ideal h pmod of projective modules and left resp right cointegrals for h the non zero left right modified traces are non degenerate and we show that non degenerate left right modified traces can only exist for unimodular h this generalises results of beliakova blanchet and gainutdinov 1 from hopf algebras to quasi hopf algebras as an example we compute cointegrals and modified traces for the family of symplectic fermion quasi hopf algebras

on modules associated to coalgebra galois extensions abstract for a given entwining structure a c involving an algebraa a coalgebrac and an entwining map c a a c a categorymca of right a c modules is defined and its structure analysed in particular the notion of a measuring of a c to a c is introduced and certain functors betweenmca andmca induced by such a measuring are defined it is shown that these functors are inverse equivalences iff they are exact or one of them faithfully exact and the measuring satisfies a certain galois type condition next left moduleseand right modules associated to ac galois extensionaofbare defined these can be thought of as objects dual to fibre bundles with coalgebracin the place of a structure group and a fibrev cross sections of such associated modules are defined as module mapse bor b it is shown that they can be identified with suitably equivariant maps from the fibre toa also it is shown that ac galois extension is cleft if and only ifa b cas leftb modules and rightc comodules the relationship between the moduleseand is studied in the case whenvis finite dimensional and in the case when the canonical entwining map is bijective

on the brauer picard group of a finite symmetric tensor category let cn denote the representation category of finite supergroup kn z 2z we compute the brauer picard group brpic cn of cn this is done by identifying brpic cn with the group of braided tensor autoequivalences of the drinfeld center of cn and studying the action of the latter group on the categorical lagrangian grassmannian of cn we show that this action corresponds to the action of a projective symplectic group on a classical lagrangian grassmannian

reflection fusion categories abstract we introduce the notion of a reflection fusion category which is a type of a g crossed category generated by objects of frobenius perron dimension 1 and p where p is an odd prime we show that such categories correspond to orthogonal reflection groups over f p this allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories

loop spaces and coherence for monoidal and braided monoidal bicategories abstract we prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories we show how coherence for these structures can be interpreted topologically using up to homotopy operad actions and the algebraic classification of surface braids

weak hopf algebras and weak yang baxter operators abstract in this paper we introduce the notions of weak yang baxter operator and weak braided hopf algebra we prove that it is possible to obtain examples of these notions working with yetter drinfeld modules associated to a weak hopf algebra h with invertible antipode finally we complete the study of the structure of weak hopf algebras with a projection obtaining a categorical equivalence between the category of weak hopf algebra projections associated to h and the category of hopf algebras in the non strict braided monoidal category of left left yetter drinfeld modules over h

effects of the reference set on frequentist inferences summary we employ second order likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function referring to this as the effect of the reference set there are two aspects of higher order corrections to first order likelihood methods namely i that involving effects of fitting nuisance parameters and leading to the modified profile likelihood and ii another part pertaining to limitation in adjusted information generally each of these involves a first order adjustment depending on the reference set however we show that for some important settings likelihood irrelevant model specifications have a second order effect on both of these adjustments this result includes specification of the censoring model for survival data on the other hand for sequential experiments the likelihood irrelevant specification of the stopping rule has a second order effect on adjustment i but a first order effect on adjustment ii these matters raise the issue of what are ideal frequentist inferences since consideration of exact frequentist inferences will not suffice we indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic without commonly considered adjustments thereto

on the central charge of a factorizable hopf algebra abstract 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

double quantum groups the construction of the drinfeld double d h of a finite dimensional hopf algebra h was one of the first examples of a quasitriangular hopf algebra whose category of modules md h is braided the braided category of yetter drinfeld modules ydhh is the analogue for infinite dimensional hopf algebras it uses a strong dependence between the h module and the h comodule structures we generalize this construction to the category mca of entwined modules that is a modules and c comodules over hopf algebras a and c where the structures are only related by an entwining map c a a c we show that mca is braided iff there is an r map r c c a a satisfying suitable axioms that generalize the axioms of an r matrix for finite dimensional c there is a quasitriangular hopf algebra structure on hom c a the quantum group double generalizing the construction of the drinfeld double

limits and colimits of hopf algebras abstract it is shown that for any commutative unital ring r the category hopf r of r hopf algebras is locally presentable and a coreflective subcategory of the category bialg r of r bialgebras admitting cofree hopf algebras over arbitrary r algebras the proofs are based on an explicit analysis of the construction of colimits of hopf algebras which generalizes an observation of takeuchi essentially be a duality argument also the dual statement namely that hopf r is closed in bialg r under limits is shown to hold provided that the ring r is von neumann regular it then follows that hopf r is reflective in bialg r and admits free hopf algebras over arbitrary r coalgebras for any von neumann regular ring r finally takeuchi s free hopf algebra construction is analysed and shown to be simply a composition of standard categorical constructions by simple dualization also a construction of the hopf coreflection is provided

asymptotic expansions in boundary crossing problems soit s n n 1 une marche aleatoire et t t x inf n 1 nf f n n on obtient des developpements asymptotiques a deux termes lorsque pour les distributions marginales de t et une somme partielle normalisee

categorical aspects of cointegrals on quasi hopf algebras abstract we discuss relations between some category theoretical notions for a finite tensor category and cointegrals on a quasi hopf algebra specifically for a finite dimensional quasi hopf algebra h we give an explicit description of categorical cointegrals of the category m h of left h modules in terms of cointegrals on h provided that h is unimodular we also express the frobenius structure of the adjoint algebra in the yetter drinfeld category y h h d by using an integral in h and a cointegral on h finally we give a description of the twisted module trace for projective h modules in terms of cointegrals on h

differential graded categories and deligne conjecture abstract we prove a version of the deligne conjecture for n fold monoidal abelian categories a over a field k of characteristic 0 assuming some compatibility and non degeneracy conditions for a the output of our construction is a weak leinster n 1 algebra over k a relaxed version of the concept of leinster n algebra in a lg k the difference between the leinster original definition and our relaxed one is apparent when n 1 for n 1 both concepts coincide we believe that there exists a functor from weak leinster n 1 algebras over k to c e n 1 k algebras well defined when k q and preserving weak equivalences for the case n 1 such a functor is constructed in 31 by elementary simplicial methods providing together with this paper a complete solution for 1 monoidal abelian categories our approach to deligne conjecture is divided into two parts the first part completed in the present paper provides a construction of a weak leinster n 1 algebra over k out of an n fold monoidal k linear abelian category provided the compatibility and non degeneracy condition are fulfilled the second part still open for n 1 is a passage from weak leinster n 1 algebras to c e n 1 k algebras as an application we prove in theorem 8 1 that the gerstenhaber schack complex of a hopf algebra over a field k of characteristic 0 admits a structure of a weak leinster 2 1 algebra over k extending the yoneda structure it relies on our earlier construction 30 of a 2 fold monoidal structure on the abelian category of tetramodules over a bialgebra

bosonization for dual quasi bialgebras and preantipode abstract to every dual quasi bialgebra h and every bialgebra r in the category of yetter drinfeld modules over h one can associate a dual quasi bialgebra called bosonization in this paper using the fundamental theorem we characterize as bosonizations the dual quasi bialgebras with a projection onto a dual quasi bialgebra with a preantipode as an application we investigate the structure of the graded coalgebra gr a associated to a dual quasi bialgebra a with the dual chevalley property e g a is pointed

generalized diagonal crossed products and smash products for quasi hopf algebras applications in this paper we introduce generalizations of diagonal crossed products two sided crossed products and two sided smash products for a quasi hopf algebra h the results we obtain may then be applied to h hopf bimodules and generalized yetter drinfeld modules the generality of our situation entails that the generating matrix formalism cannot be used forcing us to use a different approach this pays off because as an application we obtain an easy conceptual proof of an important but very technical result of hausser and nill concerning iterated two sided crossed products

on cross product hopf algebras abstract let a and b be algebras and coalgebras in a braided monoidal category c and suppose that we have a cross product algebra and a cross product coalgebra structure on a b we present necessary and sufficient conditions for a b to be a bialgebra and sufficient conditions for a b to be a hopf algebra we discuss when such a cross product hopf algebra is a double cross co product a biproduct or more generally a smash co product hopf algebra in each of these cases we provide an explicit description of the associated hopf algebra projection

group cohomology and gauge equivalence of some twisted quantum doubles the purpose of the present paper is to investigate relationships that exist between certain quasi hopf algebras namely twisted quantum doubles of a finite abelian group and their module categories this is closely related to aspects of group cohomology concerning the relationship of the bar complex to some other complexes defined and studied by eilenberg and maclane mac52 em50a and em50b although our results are purely homological and algebraic in nature the motivation for studying these questions derives mainly from connections with conformal field theory cft and vertex operator algebras dw90 dpr92 ms89 dl93 we will thus review background motivation although it is hardly necessary for an understanding of our main results and then give a more detailed overview of the paper it is not necessary for the reader to appreciate what a vertex operator algebra voa is cf bor86 flm88 fhl93 but only to understand this voas have a representation theory hence a linear module category and it is important to understand the nature of this category it is expected that the module category v mod for a voa v has the structure of a braided monoidal category js93 roughly this means that there is a notion of tensor product of v modules and that the tensor product of three modules is associative by an isomorphism that depends on the particular modules moreover tensor product of modules is commutative again via an isomorphism that depends on the particular modules particularly nice voas called rational have the additional property that v mod is a semisimple category so that v has only a finite number of isomorphism classes of simple modules and every object in v mod is a direct sum of simple objects the simplest voas of all from this perspective are called holomorphic they are rational and have but one simple module namely the adjoint module v itself thus if v is a holomorphic voa then v mod is equivalent as a semi simple braided monoidal category to the category of vector spaces

on the second fundamental theorem of invariant theory for the orthosymplectic supergroup abstract let osp v be the orthosymplectic supergroup on an orthosymplectic vector superspace v of superdimension m 2 n lehrer and zhang showed that there is a surjective algebra homomorphism f r r b r m 2 n end osp v v r where b r m 2 n is the brauer algebra of degree r with parameter m 2 n the second fundamental theorem of invariant theory in this setting seeks to describe the kernel ker f r r of f r r as a 2 sided ideal of b r m 2 n in this paper we show that ker f r r 0 if and only if r r c m 1 n 1 and give a basis and a dimension formula for ker f r r we show that ker f r r as a 2 sided ideal of b r m 2 n is generated by ker f r c r c for any r r c and we provide an explicit set of generators for ker f r c r c these generators coincide in the classical case with those obtained in recent papers of lehrer and zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups as an application we obtain the necessary and sufficient conditions for the endomorphism algebra end osp v v r over the orthosymplectic lie superalgebra osp v to be isomorphic to b r m 2 n

period in the chaotic phase of q2r automata more than 1000 h on a high speed special computer were used to analyse the global limit cycle period of the q2r automata in 2d at various energies an exponential increase of the periods with system size n was found even at energies below the curie temperature even for t to infinity i e in the fully chaotic regime the paramagnetic phase of q2r was found to be non ergodic

invariant in cellular automata for a subclass of the set of reversible cellular automata the author gives the form of an exact time invariant quantity which can be seen as a kind of energy with the ergodic assumption this is a model of two interacting ising spin systems

ribbon transformations integrals and triangular decompositions abstract we study the theory of integrals in yetter drinfel d hopf algebras and use the results to determine the integrals of hopf algebras with triangular decomposition

dynamics of interface width in the three dimensional q2r ising model the q2r cellular automaton is used to simulate the growth of the liquid vapour interface in the 3d ising model at the critical point the authors find that its width increases with time in a power law manner with an exponent of 0 345 or 0 015 below the critical temperature they study the size dependence of the equilibrium width

numerical method for analyzing surface fluctuations abstract a new method for measuring the width of an interface in the presence of overhangs and holes is presented using a local gibbs dividing surface it is compared with the conventional method based on the order parameter profile dynamics and size dependence of an interface are studied numerically in the two dimensional q2r cellular automaton two types of interface fluctuations can be distinguished the capillary waves dominate on length scales larger than the bulk correlation length whereas order parameter fluctuations on small scales give rise to an intrinsic interface width the scaling behavior of the capillary waves is shown to be the same as for the ising model the divergence of the intrinsic width at the critical point and its critical slowing down are investigated

butterflies i morphisms of 2 group stacks abstract weak morphisms of non abelian complexes of length 2 or crossed modules are morphisms of the associated 2 group stacks or gr stacks we present a full description of the weak morphisms in terms of diagrams we call butterflies we give a complete description of the resulting bicategory of crossed modules which we show is fibered and biequivalent to the 2 stack of 2 group stacks as a consequence we obtain a complete characterization of the non abelian derived category of complexes of length 2 deligne s analogous theorem in the case of picard stacks and abelian sheaves becomes an immediate corollary commutativity laws on 2 group stacks are also analyzed in terms of butterflies yielding new characterizations of braided symmetric and picard 2 group stacks furthermore the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non abelian cohomology removing a preexisting fibration condition on the coefficients short exact sequence

monoidal 2 structure of bimodule categories abstract we define a notion of tensor product of bimodule categories and prove that with this product the 2 category of c bimodule categories for fixed tensor c is a monoidal 2 category in the sense of kapranov and voevodsky 1991 1 we then provide a monoidal structure preserving 2 equivalence between the 2 category of c bimodule categories and z c module categories module categories over the center of c for a finite group g we show that de equivariantization is equivalent to the tensor product over rep g we derive rep g module fusion rules and show that the group of invertible rep g module categories is isomorphic to h 2 g k extending results in etingof et al 2

a modular invariant bulk theory for the c 0 triplet model a proposal for the bulk space of the logarithmic cal w 2 3 triplet model at central charge zero is made the construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory the resulting bulk space is a quotient of the direct sum of projective representations which is isomorphic as a vector space to the direct sum of tensor products of the irreducible representations with their projective covers as a consistency check of our analysis we show that the partition function of the bulk theory is modular invariant and that the boundary state analysis is compatible with the proposed annulus partition functions of this model

hopf modules and the double of a quasi hopf algebra we give a different proof for a structure theorem of hausser and nill on hopf modules over quasi hopf algebras we extend the structure theorem to a classification of two sided two cosided hopf modules by yetter drinfeld modules which can be defined in two rather different manners for the quasi hopf case the category equivalence between hopf modules and yetter drinfeld modules leads to a new construction of the drinfeld double of a quasi hopf algebra as proposed by majid and constructed by hausser and nill

actions of monoidal categories and generalized hopf smash products let r be a k algebra and c a monoidal category assume given the structure of a c category on the category rm of left r modules that is the monoidal category c is assumed to act on the category rm by a coherently associative bifunctor c rm rm we assume that this bifunctor is right exact in its right argument in this setup we show that every algebra a respectively coalgebra c in c gives rise to an r ring a r respectively an r coring c r whose modules respectively comodules are the a modules respectively c comodules within the category rm we show that this very general scheme for constructing co associative co rings gives conceptual explanations for the double of a quasi hopf algebra as well as certain doubles of hopf algebras in braided categories each time avoiding ad hoc computations showing associativity

doubles of quasi quantum groups in dr1 drinfeld showed that any finite dimensional hopf algebra extends to a quasitriangular hopf algebra the quantum double of based on the construction of a so called diagonal crossed product developed by the authors in hn we generalize this result to the case of quasi hopf algebras as for ordinary hopf algebras as a vector space the quasi quantum double is isomorphic to where denotes the dual of we give explicit formulas for the product the coproduct the r matrix and the antipode on and prove that they fulfill drinfeld s axioms of a quasitriangular quasi hopf algebra in particular becomes an associative algebra containing as a quasi hopf subalgebra on the other hand is not a subalgebra of unless the coproduct on is strictly coassociative it is shown that the category of finite dimensional representations of coincides with what has been called the double category of modules by s majid m2 thus our construction gives a concrete realization of majid s abstract definition of quasi quantum doubles in terms of a tannaka krein like reconstruction procedure the whole construction is shown to generalize to weak quasi hopf algebras with now being linearly isomorphic to a subspace of

dg analogues of the zuckerman functors and the dual zuckerman functors i abstract we study the category of dg harish chandra modules over arbitrary commutative rings and generalize the induction functor the production functor the zuckerman functor and the dual zuckerman functor

frobenius schur indicators and exponents of spherical categories we obtain two formulae for the higher frobenius schur indicators one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist the first one is a categorical generalization of an analogous result by kashina sommerhauser and zhu for hopf algebras and the second one extends bantay s 2nd indicator formula for a conformal field theory to higher degrees these formulae imply the sequence of higher indicators of an object in these categories is periodic we define the notion of frobenius schur fs exponent of a pivotal category to be the global period of all these sequences of higher indicators and we prove that the fs exponent of a spherical fusion category is equal to the order of the twist of its center consequently the fs exponent of a spherical fusion category is a multiple of its exponent in the sense of etingof by a factor not greater than 2 as applications of these results we prove that the exponent and the dimension of a semisimple quasi hopf algebra h have the same prime divisors which answers two questions of etingof and gelaki affirmatively for quasi hopf algebras moreover we prove that the fs exponent of h divides dim h 4 in addition if h is a group theoretic quasi hopf algebra the fs exponent of h divides dim h 2 and this upper bound is shown to be tight

homological epimorphisms recollements and hochschild cohomology with a conjecture by snashall solberg in view abstract we show that recollements of module categories give rise to homomorphisms between the associated hochschild cohomology algebras which preserve the strict gerstenhaber structure i e the cup product the graded lie bracket and the squaring map we review various long exact sequences in hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict gerstenhaber structure as well as a byproduct we generalise a known long exact cohomology sequence of koenig nagase to arbitrary surjective homological epimorphisms we use our observations to motivate and formulate a variation of the finite generation conjecture by snashall solberg

infinite loop spaces and coherence for symmetric monoidal bicategories abstract this paper proves three different coherence theorems for symmetric monoidal bicategories first we show that in a free symmetric monoidal bicategory every diagram of 2 cells commutes second we show that this implies that the free symmetric monoidal bicategory on one object is equivalent as a symmetric monoidal bicategory to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups third we show that every symmetric monoidal bicategory is equivalent to a strict one we give two topological applications of these coherence results first we show that the classifying space of a symmetric monoidal bicategory can be equipped with an e structure second we show that the fundamental 2 groupoid of an e n space n 4 has a symmetric monoidal structure these calculations also show that the fundamental 2 groupoid of an e 3 space has a sylleptic monoidal structure

extensions of q2r potts model and other lattices the authors extend the q2r rule from the ising to the potts model and also to random lattices remedies against the lack of unique ergodicity of q2r are also presented