measurable kac cohomology for bicrossed products we study the kac cohomology for matched pairs of locally compact groups this cohomology theory arises from the extension theory of locally compact quantum groups we prove a measurable version of the kac exact sequence and provide methods to compute the cohomology we give explicit calculations in several examples using results of moore and wigner

on group theoretical hopf algebras and exact factorizations of finite groups abstract we show that a semisimple hopf algebra a is group theoretical if and only if its drinfeld double is a twisting of the dijkgraaf pasquier roche quasi hopf algebra d for some finite group and some z 3 k we show that semisimple hopf algebras obtained as bicrossed products from an exact factorization of a finite group are group theoretical we also describe their drinfeld double as a twisting of d for an appropriate 3 cocycle coming from the kac exact sequence

extending lazy 2 cocycles on hopf algebras and lifting projective representations afforded by them abstract we study some problems related to lazy 2 cocycles such as extension of lazy 2 cocycles to a drinfeld double and to a radford biproduct yetter drinfeld data obtained from lazy 2 cocycles lifting of projective representations afforded by lazy 2 cocycles

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

on bisimulations for description logics we study bisimulations for useful description logics the simplest among the considered logics is alc reg a variant of pdl the others extend that logic with the features inverse roles nominals qualified number restrictions the universal role and or the concept constructor for expressing the local reflexivity of a role they also allow role axioms our contributions are as follows we propose to treat named individuals as initial states and give an appropriate bisimulation condition for that we also give bisimulation conditions for the universal role and the concept constructor r self we prove that all of the bisimulation conditions for the features can be combined together to guarantee invariance of concepts and the hennessy milner property for the whole class of studied description logics we address and give results on invariance or preservation of aboxes rboxes and knowledge bases in description logics independently from lutz et al 26 27 we also give results on invariance of tboxes we introduce a new notion called qs interpretation which is needed for dealing with minimizing interpretations in description logics with qualified number restrictions and or the concept constructor r self we formulate and prove results on minimality of quotient interpretations w r t the largest auto bisimulations we adapt hopcroft s automaton minimization algorithm to give an efficient algorithm for computing the partition corresponding to the largest auto bisimulation of a finite interpretation in any description logic of the considered family using the invariance results we compare the expressiveness of the considered description logics w r t concepts tboxes and aboxes our results about separating the expressiveness of description logics are naturally extended to the case when instead of alc reg we have any sublogic of alc reg that extends alc

the lazy homology of a hopf algebra to any hopf algebra h we associate two commutative hopf algebras h1l h and h2l h which we call the lazy homology hopf algebras of h these hopf algebras are built in such a way that they are predual to the lazy cohomology groups based on the so called lazy cocycles we establish two universal coefficient theorems relating the lazy cohomology to the lazy homology and we compute the lazy homology of the sweedler algebra

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

on marcinkiewicz zygmund laws abstract marcinkiewicz zygmund laws with convergence rates are established here for a class of strictly stationary ergodic sequences

the brauer group of modified supergroup algebras abstract the computation of the brauer group bm of modified supergroup algebras over a field is performed yielding in particular the computation of the brauer group of all finite dimensional triangular hopf algebras when the base field is algebraically closed and of characteristic zero the used techniques are also applied for the computation of lazy cohomology of modified supergroup algebras we relate our construction with yinhuo zhang s exact sequence when the base ring is a field as an example we compute explicitly the brauer group and lazy cohomology for modified supergroup algebras with extensions of weyl groups of irreducible root systems as a group datum and their standard representation as a representation datum

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

a decomposition of the brauer picard group of the representation category of a finite group abstract we present an approach of calculating the group of braided autoequivalences of the category of representations of the drinfeld double of a finite dimensional hopf algebra h and thus the brauer picard group of h mod we consider two natural subgroups and a subset as candidates for generators in this article h is the group algebra of a finite group g as our main result we prove that any element of the brauer picard group fulfilling an additional cohomological condition decomposes into an ordered product of our candidates for elementary abelian groups g our decomposition reduces to the bruhat decomposition of the brauer picard group which is in this case a lie group over a finite field our results are motivated by and have applications to symmetries and defects in 3d tqft and group extensions of fusion categories

cross product bialgebras i abstract the subject of this article is bialgebra factorizations or cross product bialgebras without cocycles we establish a theory characterizing cross product bialgebras universally in terms of projections and injections especially all known types of biproduct double cross product and bicross product bialgebras can be described by this theory furthermore the theory provides new families of cocycle free cross product bialgebras besides the universal characterization we find an equivalent co modular description of certain types of cross product bialgebras in terms of so called hopf data with the help of hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras we are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of hopf bimodules over a hopf algebra majid s double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category this resembles the case of the drinfel d double which can be constructed as a twist of a specific cross product

extensions of hopf algebras and lie bialgebras let f g be finite dimensional lie algebras over a field of characteristic zero regard f and g the dual lie coalgebra of g as lie bialgebras with zero cobracket and zero bracket respectively suppose that a matched pair f g of lie bialgebras is given which has structure maps then it induces a matched pair u f ug of hopf algebras where u f is the universal envelope of f and ug is the hopf dual of ug we show that the group opext u f ug of cleft hopf algebra extensions associated with u f ug is naturally isomorphic to the group opext f g of lie bialgebra extensions associated with f g an exact sequence involving either of these groups is obtained which is a variation of the exact sequence due to g i kac if g g g there follows a bijection between the set ext u f ug of all cleft hopf algebra extensions of u f by ug and the set ext f g of all lie bialgebra extensions of f by g

braided bi galois theory ii the cocommutative case abstract we study groups of bi galois objects over a hopf algebra h in a braided monoidal category b we assume h to be cocommutative in a certain sense this does not mean that h is a cocommutative coalgebra with respect to the braiding given in b but it is cocommutative with respect to a different braiding subject to specific axioms the type of cocommutative hopf algebras under consideration investigated in previous papers occurs naturally for example in majid s transmutation construction we show that for cocommutative h the suitably defined cocommutative bi galois objects form a subgroup in the group of h h bi galois objects we also show that all cocycles on h are lazy and that second lazy cohomology describes the subgroup of cleft bi galois extensions in the group of cocommutative ones

on the gauge equivalence of twisted quantum doubles of elementary abelian and extra special 2 groups abstract we establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group g by an abelian group with 3 cocycle inflated from a 3 cocycle on g we also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences we give two main applications first if g is an extra special 2 group of width at least 2 we show that the quantum double of g twisted by a 3 cocycle is gauge equivalent to a twisted quantum double of an elementary abelian 2 group if and only if 2 is trivial second we discuss the gauge equivalence classes of twisted quantum doubles of groups of order 8 and classify the braided tensor equivalence classes of these quasi triangular quasi bialgebras it turns out that there are exactly 20 such equivalence classes

cohomology and coquasi bialgebra extensions associated to a matched pair of bialgebras abstract by using braid diagrams we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras in order to give a method of constructing coquasi bialgebras which generalize bialgebras and classifying them up to monoidal equivalence of their comodule categories an alternative homological proof is given for schauenburg s generalized kac sequence involving the abelian group opext h k of bialgebra extensions we define an abelian group opext h k of coquasi bialgebra extensions associated to a singer pair h k of bialgebras and prove a variant of schauenburg s sequence which involves the group it is also proved that there is a natural isomorphism opext h 1 k opext h 2 k that preserves monoidal equivalence classes if h 1 k and h 2 k arise from such matched pairs that are shifts of each other

exact factorizations and extensions of fusion categories abstract we introduce and study the new notion of an exact factorization b a c of a fusion category b into a product of two fusion subcategories a c b of b this is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups we then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category which was introduced and studied in 7 we also apply our results to study extensions of a group theoretical fusion category by another one provide some examples and propose a few natural questions

calculations of some groups of hopf algebra extensions w let us recall the theory of group extensions with abelian kernels mc x chap iv let g be a group and m a g module the equivalence classes of group extensions m a e a g giving rise to the g module action on m form a group with respect to the baer sum furthermore this group is 2 isomorphic to the cohomology group h g m the theory is extended to the theory of hopf algebra extensions which w x are abelian in some sense h s we restrict our interests to hopf algebra extensions over a field k which are of the form

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

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

maximal inequalities for partial sums of rho mixing sequences a rosenthal type inequality for maximum partial sums of p mixing sequences is obtained applications to the complete convergence and almost sure summability of partial sums are also discussed

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h