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

frobenius schur indicators in tambara yamagami categories abstract we introduce formulae of frobenius schur indicators of simple objects of tambara yamagami categories by using techniques of the fourier transform on finite abelian groups we study some arithmetic properties of indicators we also give some applications to hopf algebras

the green rings of the 2 rank taft algebra and its two relatives twisted abstract in the paper the representation rings or the green rings for a family of hopf algebras of tame type the 2 rank taft algebra at q 1 and its two relatives twisted by 2 cocycles are explicitly described via a representation theoretic analysis it turns out that the green rings can serve to detect effectively the twist equivalent hopf algebras here

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

classification of metaplectic modular categories abstract we obtain a classification of metaplectic modular categories every metaplectic modular category is a gauging of the particle hole symmetry of a cyclic modular category our classification suggests a conjecture that every weakly integral modular category can be obtained by gauging a symmetry including the fermion parity of a pointed super modular category

scaling behavior of stochastic minimization algorithms in a perfect funnel landscape we determined scaling laws for the numerical effort to find the optimal configurations of a simple model potential energy surface pes with a perfect funnel structure that reflects key characteristics of the protein interactions generalized monte carlo methods mcm stun avoid an enumerative search of the pes and thus provide a natural resolution of the levinthal paradox we find that the computational effort grows with approximately the eighth power of the system size for mcm and stun while a genetic algorithm was found to scale exponentially the scaling behaviour of a derived lattice model is also rationalized

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

on semisimple hopf algebras of dimension 2q3 abstract let q be a prime number k an algebraically closed field of characteristic 0 and h a semisimple hopf algebra of dimension 2 q 3 this paper proves that h is always semisolvable that is such hopf algebras can be obtained by a number of extensions from group algebras or duals of group algebras

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 slightly degenerate fusion categories abstract in this paper we first show for a slightly degenerate pre modular fusion category c that squares of dimensions of simple objects divide half of the dimension of c and that slightly degenerate fusion categories of fp dimensions 2 p n d and 4 p n d are nilpotent where p is an odd prime and d is an odd square free integer then we classify weakly integral slightly degenerate fusion categories of particular dimensions

efficient perturbation analysis of elastic network models application to acetylcholinesterase of t californica elastic network models in their different flavors have become useful models for the dynamics and functions of biomolecular systems such as proteins and their complexes perturbation to the interactions occur due to randomized and fixated changes in molecular evolution or designed modifications of the protein structures in bioengineering these perturbations are modifications in the topology and the strength of the interactions modeled by the elastic network models we discuss how a naive approach to compute properties for a large number of perturbed structures and interactions by repeated diagonalization can be replaced with an identity found in linear algebra we argue about the computational complexity and discuss the advantages of the protocol we apply the proposed algorithm to the acetylcholinesterase a well known enzyme in neurobiology and show how one can gain insight into the breathing dynamics of a structural funnel necessary for the function of the protein the computational speed up was a 60 fold increase in this example

a jordan holder theorem for weakly group theoretical fusion categories we prove a version of the jordan holder theorem in the context of weakly group theoretical fusion categories this allows us to introduce the composition factors and the length of such a fusion category mathcal c which are in fact morita invariants of mathcal c

a family of isomorphic fusion algebras of twisted quantum doubles of finite groups let d g be the twisted quantum double of a finite group g where z3 g c for each n n there exists an such that d g and d e have isomorphic fusion algebras where g is an extraspecial 2 group with 22n 1 elements and e is an elementary abelian group with e g

lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups we classify lagrangian subcategories of the representation category of a twisted quantum double d g where g is a finite group and is a 3 cocycle on it in view of results of dgno this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups we also establish a canonical bijection between lagrangian subcategories of rep d g and module categories over the category rm vec g omega of twisted g graded vector spaces such that the dual tensor category is pointed this can be viewed as a quantum version of v drinfeld s characterization of homogeneous spaces of a poisson lie group in terms of lagrangian subalgebras of the double of its lie bialgebra d as a consequence we obtain that two group theoretical fusion categories are weakly morita equivalent if and only if their centers are equivalent as braided tensor categories

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 a symmetry of muger s centralizer for the drinfeld double of a semisimple hopf algebra in this paper we prove a formula that relates muger s centralizer in the category of representations of a factorizable hopf algebra to the notion of hopf kernel of a representation of the dual hopf algebra using this relation we obtain a complete description for muger s centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a drinfeld double of a semisimple hopf algebra

on the brauer picard groups of fusion categories we develop methods of computation of the brauer picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension representation categories of elementary abelian groups with twisted associativity extra special p groups and the kac paljutkin hopf algebra we conclude that many finite groups of lie type occur as composition factors of the brauer picard groups of pointed fusion categories

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

on an integral transform with kernel o_ v 1 2 mu cos theta and its applications to second order canonical problems of gtd an arbitrary function defined for almost every theta in infty infty can be represented under certain restrictions by an integral with kernel o v 1 2 mu cos theta this representation has some analytical properties similar to fourier integrals one of the basic postulates of gtd the geometrical theory of diffraction which states high frequency diffraction as a local phenomenon makes this representation effectively suitable to solve second order canonical problems of gtd related to edges on spherical surfaces as an illustrative example the diffraction of a whispering gallery mode by a spherical cap is studied

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

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

finite group factorizations and braiding we compute the quantum double braiding and other canonical hopf algebra constructions for the bicrossproduct hopf algebrahassociated to the factorization of a finite groupxinto two subgroups the representations of the quantum double are described by a notion of bicrossed bimodules generalising the cross modules of whitehead we also show that basis preserving self duality structures for the bicrossproduct hopf algebras are in one one correspondence with factor reversing group isomorphisms the example z6z6is given in detail we show further that the quantum doubled h is the twisting ofd x by a nontrivial quantum cocycle

transition functions for high frequency diffraction by a curved perfectly conducting wedge i canonical solution for a curved sheet starting from the canonical hilbert solution for a perfectly conducting cylindrically curved sheet of semi infinite angular extent illuminated by a line source transition functions are derived for illumination and or observation directions grazing the sheet surface or its geometrical continuation near the edge this is done by using the langer olver asymptotic formulas for certain hankel and bessel function factors in the hilbert solution both the soft transverse magnetic and the hard transverse electric cases are considered depending on the positions of the source and or the observer the resulting functions include the fock functions the incomplete airy function and its derivative and a pair of new canonical functions the results obtained can serve as a check for heuristic solutions for a curved wedge

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

weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

nilpotent fusion categories abstract in this paper we extend categorically the notion of a finite nilpotent group to fusion categories to this end we first analyze the trivial component of the universal grading of a fusion category c and then introduce the upper central series of c for fusion categories with commutative grothendieck rings e g braided fusion categories we also introduce the lower central series we study arithmetic and structural properties of nilpotent fusion categories and apply our theory to modular categories and to semisimple hopf algebras in particular we show that in the modular case the two central series are centralizers of each other in the sense of m muger

z 2z extensions of hopf algebra module categories by their base categories abstract starting with a self dual hopf algebra h in a braided monoidal category s we construct a z 2 z graded monoidal category c c 0 c 1 the degree zero component is the category rep s h of representations of h and the degree one component is the category s the extra structure on h needed to define the associativity isomorphisms is a choice of self duality map and cointegral subject to certain conditions we also describe rigid braided and ribbon structures on c in hopf algebraic terms our construction permits a uniform treatment of tambara yamagami categories and categories related to symplectic fermions in conformal field theory

pseudo unitary non self dual fusion categories of rank 4 abstract a fusion category of rank 4 has either four self dual simple objects or exactly two self dual simple objects we study fusion categories of rank 4 with exactly two self dual simple objects giving nearly a complete classification of those based rings that admit pseudo unitary categorification more precisely we show that if c is such a fusion category then its grothendieck ring k c must be one of seven based rings six of which have known categorifications in doing so we classify all based rings associated with near group categories of the group z 3 z

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

an explicit fusion algebra isomorphism for twisted quantum doubles of finite groups we exhibit an isomorphism between the fusion algebra of the quantum double of an extraspecial p group where p is an odd prime and the fusion algebra of a twisted quantum double of an elementary abelian group of the same order

central invariants and frobenius schur indicators for semisimple quasi hopf algebras abstract in this paper we obtain a canonical central element h for each semi simple quasi hopf algebra h over any field k and prove that h is invariant under gauge transformations we show that if k is algebraically closed of characteristic zero then for any irreducible representation of h which affords the character h takes only the values 0 1 or 1 moreover if h is a hopf algebra or a twisted quantum double of a finite group then h is the corresponding frobenius schur indicator we also prove an analog of a theorem of larson radford for split semi simple quasi hopf algebras over any field k using this result we establish the relationship between the antipode s the values of h and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero

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

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

energy landscape paving as a perfect optimization approach under detrended fluctuation analysis abstract global optimization go is one of the key numerical tools in computational physics among the go algorithms the ones originating in statistical physics are particularly powerful recently an adaptive scheme was developed to increase the efficiency of one of these algorithms stochastic tunneling this scheme is based on the time series of minima tested and the respective detrended fluctuation analysis dfa we here present a study on another go methodology energy landscape paving which in itself is adaptive and show that its performance is optimal under the dfa analysis we give arguments to explain this fact

classifying weak coideal subalgebras of weak hopf c algebras abstract we develop a general approach to the problem of classification of weak coideal c subalgebras of weak hopf c algebras as an example we consider weak hopf c algebras and their weak coideal c subalgebras associated with tambara yamagami categories

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

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

quantum lines over non cocommutative cosemisimple hopf algebras the recognition of the importance of braided tensor categories came by the remarkable recent developments of quantum group theory an especially important example of such a category is the braided tensor category hyd of yetter drinfeld modules over a hopf algebra h with bijective antipode it is interesting and delightful to see that old results on ordinary hopf algebras such as those by nichols 18 1978 and by radford 19 1985 are reproduced in terms of the new notion of braided hopf algebra in hyd given a braided hopf algebra r in hyd one can construct an ordinary hopf algebra r h of biproduct this naturally forms a triple r h called a hopf algebra triple over h such that h r h and r h h are hopf algebra maps with id if we define h 1 h x h e x h it was essentially proved by radford 19 and then reproduced by majid 13 that r r h gives a category equivalence called bosonisation from the category of braided hopf algebras in hhyd to the category of hopf algebra triples over h see proposition 1 1

fusion rings with few degrees abstract the main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p a well known theorem of isaacs and passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary numerically compatible nc fusion rings are defined as a potential generalization of the grothendieck rings of fusion categories and the nc fusion rings of order p p 2 p 3 pq and p q 2 where p q are primes are classified in terms of certain standard table algebras as another application of the main theorem

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

finite quantum groups and quantum permutation groups abstract we give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups we also give examples of finite quantum groups which are not quantum permutation groups one such example occurs as a split abelian extension associated to the exact factorization s 4 z 4 s 3 and has dimension 24 we show that in fact this is the smallest possible dimension that a non quantum permutation group can have

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

near group fusion categories and their doubles a near group fusion category is a fusion category c where all but 1 simple objects are invertible examples of these include the tambara yamagami categories and the even sectors of the e6 and affine d5 subfactors though there are infinitely many others we classify the near group fusion categories and compute their doubles and the modular data relevant to conformal field theory among other things we explicitly construct over 40 new finite depth subfactors with jones index ranging from around 6 85 to around 14 93 we expect all of these doubles to be realised by rational conformal field theories

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

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

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

diffraction of a whispering gallery mode by the edge of a thin concave cylindrically curved surface when a cylindrically curved perfectly conducting concave surface is terminated abruptly an incident whispering gallery wg mode undergoes diffraction this phenomenon is studied here for the canonical problem of a thin boundary and an edge that is parallel to the cylinder axis to confine attention to diffraction by a single edge it is assumed that the boundary extends to infinity in an infinitely extended angular space that is equivalent to placing along some radial plane intersecting the boundary a perfect absorber for angularly propagating waves the problem can then be phrased as a functional equation problem of the hilbert type which is solved exactly an asymptotic approximation in the high frequency limit ka gg 1 where k is the wavelength and a the radius of curvature is obtained in explicit form and phrased so as to exhibit the diffraction coefficient for edge diffracted rays and the launching coefficients for creeping waves on the convex side as well as wg modes on the concave side these quantities are of interest for placing the results within the context of the geometrical theory of diffraction gtd and are relevant to the analysis of curved two dimensional reflectors of finite width where wg modes excited by diffraction of the incident field at one edge provide illumination of the other edge the induced surface current density function is also derived the asymptotic results including an observed shift of the shadow boundary relative to the direction of incidence of the wg mode are explained completely in terms of half plane diffraction of one of the modal ray congruences thereby confirming via this canonical problem the validity of the gtd for diffraction of modal fields for the limiting case a rightarrow infty all expressions are shown to reduce to those for the classical half plane problem

adaptive extremal optimization by detrended fluctuation analysis global optimization is one of the key challenges in computational physics as several problems e g protein structure prediction the low energy landscape of atomic clusters detection of community structures in networks or model parameter fitting can be formulated as global optimization problems extremal optimization eo has become in recent years one particular successful approach to the global optimization problem as with almost all other global optimization approaches eo is driven by an internal dynamics that depends crucially on one or more parameters recently the existence of an optimal scheme for this internal parameter of eo was proven so as to maximize the performance of the algorithm however this proof was not constructive that is one cannot use it to deduce the optimal parameter itself a priori in this study we analyze the dynamics of eo for a test problem spin glasses based on the results we propose an online measure of the performance of eo and a way to use this insight to reformulate the eo algorithm in order to construct optimal values of the internal parameter online without any input by the user this approach will ultimately allow us to make eo parameter free and thus its application in general global optimization problems much more efficient

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

representations of the loop braid groups from braided tensor categories the loop braid group is the motion group of unknotted oriented circles in r3 in this paper we study their representations through the approach inspired by two dimensional topological phases of matter in principle the motion of loops in r3 reduces to the motions of points in a two dimensional sliced plane we realize this physical picture in terms of braided tensor categories and their braid group representations

tensor categories with fusion rules of self duality for finite abelian groups abstract semisimple tensor categories with fusion rules of self duality for finite abelian groups are classified as an application we prove that the tannaka duals of the dihedral and the quaternion groups of order 8 and the eight dimensional hopf algebra of kac and paljutkin are not isomorphic as abstract tensor categories whereas they have the same fusion rule

diffraction of the creeping waves generated on a perfectly conducting spherical scatterer by a ring source to analyze the diffraction of high frequency waves one turns often to the geometrical theory of diffraction gtd whose aim is to describe this phenomenon in terms of certain factors these factors involve among the others several diffraction coefficients showing the modifications to be considered when a ray is transformed into another one at a diffraction point the aim of this paper is to analyze the diffraction of creeping waves generated on a perfectly conducting spherical reflector and thereby to obtain explicit expressions for certain coefficients related to the diffractions occurring at the edges of spherically curved reflectors the analysis is performed by using an integral transform technique recently developed by one of the authors various ray contributions are isolated and fairly simple nonuniform expressions for the diffraction coefficients are obtained

tannaka duals in semisimple tensor categories abstract tannaka duals of finite dimensional 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 monoidally equivalent to the initial tensor categories in a canonical fashion

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

central invariants and higher indicators for semisimple quasi hopf algebras in this paper we define the higher frobenius schur fs indicators for finite dimensional modules of a semisimple quasi hopf algebra h via the categorical counterpart developed in a 2005 preprint when h is an ordinary hopf algebra we show that our definition coincides with that introduced by kashina sommerhauser and zhu we find a sequence of gauge invariant central elements of h such that the higher fs indicators of a module v are obtained by applying its character to these elements as an application we show that fs indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by tambara and yamagami three of these categories correspond to well known hopf algebras and we explicitly construct a quasi hopf algebra corresponding to the fourth one using the kac algebra we also derive explicit formulae for fs indicators for some quasi hopf algebras associated to group cocycles