latin hypercube designs with controlled correlations and multi dimensional stratification various methods have been proposed to construct latin hypercube designs with small correlations orthogonal arrays have been used to construct latin hypercube designs with multi dimensional stratification to integrate these two ideas we propose a method to construct latin hypercube designs with both controlled correlations and multi dimensional stratification for numerical integration the constructed designs not only filter out lower dimensional variance components as effectively as ordinary orthogonal array based latin hypercube designs but also filter out bilinear terms more effectively the proposed construction method entails no iterative searches sampling properties of the constructed designs are derived examples are given to illustrate the proposed construction method and the theoretical results

variable selection for gaussian process models in computer experiments in many situations simulation of complex phenomena requires a large number of inputs and is computationally expensive identifying the inputs that most impact the system so that these factors can be further investigated can be a critical step in the scientific endeavor in computer experiments it is common to use a gaussian spatial process to model the output of the simulator in this article we introduce a new simple method for identifying active factors in computer screening experiments the approach is bayesian and only requires the generation of a new inert variable in the analysis however in the spirit of frequentist hypothesis testing the posterior distribution of the inert factor is used as a reference distribution against which the importance of the experimental factors can be assessed the methodology is demonstrated on an application in material science a computer experiment from the literature and simulated examples

group actions on central simple algebras abstract let g be a group f a field and a a finite dimensional central simple algebra over f on which g acts by f algebra automorphisms we study the subalgebras and ideals of a which are preserved by the group action we prove a structure theorem and two classification theorems for invariant subalgebras under suitable hypotheses on a we illustrate these results in the case of compact connected lie groups and give some other applications we also classify invariant ideals

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

congruence subgroups and generalized frobenius schur indicators we introduce generalized frobenius schur indicators for pivotal categories in a spherical fusion category mathcal c an equivariant indicator of an object in mathcal c is defined as a functional on the grothendieck algebra of the quantum double z mathcal c via generalized frobenius schur indicators the set of all equivariant indicators admits a natural action of the modular group using the properties of equivariant indicators we prove a congruence subgroup theorem for modular categories as a consequence all modular representations of a modular category have finite images and they satisfy a conjecture of eholzer in addition we obtain two formulae for the generalized indicators one of them a generalization of bantay s second indicator formula for a rational conformal field theory this formula implies a conjecture of pradisi sagnotti stanev as well as a conjecture of borisov halpern schweigert

twists of quantum borel algebras abstract we classify drinfeld twists for the quantum borel subalgebra u q b in the frobenius lusztig kernel u q g where g is a simple lie algebra over c and q an odd root of unity more specifically we show that alternating forms on the character group of the group of grouplikes for u q b generate all twists for u q b under a certain algebraic group action this implies a simple classification of finite dimensional hopf algebras whose categories of representations are tensor equivalent to that of u q b we also show that cocycle twists for the corresponding de concini kac algebra are in bijection with alternating forms on the aforementioned character group

minimum aberration designs with two level and four level factors summary for optimal assignment of factors to columns of a 4 2 n design we consider extensions of the minimum aberration criterion for the 2 level designs minimum aberration 412n and 422n designs in 16 and 32 runs are obtained and tabulated some results on the wordlength patterns and design classification are obtained

mappin class group actions on quantum doubles we study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non semisimple hopf algebra that arise in the construction of universal extended topological field theories we discuss how for doubles the degeneracy problem of tqft s is circumvented we find compact formulae for thes 1 matrices using the canonical non degenerate forms of hopf algebras and the bicrossed structure of doubles rather than monodromy matrices a rigorous proof of the modular relations and the computation of the projective phases is supplied using radford s relations between the canonical forms and the moduli of integrals we analyze the projectivesl 2 z action on the center ofuq sl2 forq anl 2m 1st root of unity it appears that the 3m 1 dimensional representation decomposes into anm 1 dimensional finite representation and a2m dimensional irreducible representation the latter is the tensor product of the two dimensional standard representation ofsl 2 z and the finite m dimensional representation obtained from the truncated tqft of the semisimplified representation category ofuq sl2

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

the character table for a hopf algebra arising from the drinfel d double abstract in this paper we give a definition of the character table for a hopf algebra h over a commutative field k of characteristic 0 when h is a finite dimensional semisimple hopf algebra over an algebraically closed field k of characteristic 0 we obtain a formula for the character table which extends the notion of character table for a finite group

hopf algebras of low dimension the main aim of this paper is to classify all types of hopf algebras of dimension less thn or equal to 11 over an algebraically closed field of characteristic 0 if a is such a hopf algebra that is not semisimple then we shall prove that a or a is pointed this property will result from the fact that under some assumptions any hopf algebra that is generated as an algebra by a four dimensional simple subcoalgebra is a hopf quotient of the coordinate ring of quantum sl 2 k the first result allows us to reduce the classification to the case of pointed hopf algebras of dimension 8 we shall describe their types in the last part of the paper

balanced repeated replications based on orthogonal multi arrays summary to obtain variance estimates in stratified sampling with possibly unequal numbers of primary sampling units per stratum one can either randomly divide the units in each stratum into two groups and apply the balanced half samples method to these groups or use mixed orthogonal arrays to form a set of balanced subsamples with the first technique resulting variance estimates can be very inefficient and with the second technique it can be difficult to construct orthogonal arrays with a small number of replicates for a given stratified design the proposed extension of the orthogonal array method to orthogonal multi arrays retains all of the orthogonal array method s desirable properties but allows greater flexibility in construction and thus can handle more complex stratified designs using fewer replicates

blocking schemes for definitive screening designs in earlier work jones and nachtsheim proposed a new class of screening designs called definitive screening designs as originally presented these designs are three level designs for quantitative factors that provide estimates of main effects that are unbiased by any second order effect and require only one more than twice as many runs as there are factors definitive screening designs avoid direct confounding of any pair of second order effects and for designs that have more than five factors project to efficient response surface designs for any two or three factors recently jones and nachtsheim expanded the applicability of these designs by showing how to include any number of two level categorical factors however methods for blocking definitive screening designs have not been addressed in this article we develop orthogonal blocking schemes for definitive screening designs we separately consider the cases where all of the factors are quantitative and where there is a mix of quantitative and tw

exact relations for effective tensors of polycrystals ii applications to elasticity and piezoelectricity an important necessary condition for an exact relation for effective moduli of polycrystals to hold is stability of that relation under lamination this requirement is so restrictive that it is possible if not always feasible to find all such relations explicitly in order to do this one needs to combine the results developed in part i of this paper and the representation theory of the rotation groups so 2 and so 3 more precisely one needs to know all rotationally invariant subspaces of the space of material moduli this paper presents an algorithm for finding all such subspaces we illustrate the workings of the algorithm on the examples of 3 dimensional elasticity where we get all the exact relations and the examples of 2 dimensional and 3 dimensional piezoelectricity where we get some possibly all of them

hopf algebras of dimension pq ii abstract let h be a hopf algebra of dimension pq over an algebraically closed field of characteristic zero where p q are odd primes with p q 4 p 11 we prove that h is semisimple and thus isomorphic to a group algebra or the dual of a group algebra

a bayesian variable selection approach for analyzing designed experiments with complex aliasing experiments using designs with complex aliasing patterns are often performed for example twolevel nongeometric plackett burman designs multilevel and mixed level fractional factorial designs two level fractional factorial designs with hard to control factors and supersaturated designs hamada and wu proposed an iterative guided stepwise regression strategy for analyzing the data from such designs that allows entertainment of interactions their strategy provides a restricted search in a rather large model space however this article provides an efficient methodology based on a bayesian variable selection algorithm for searching the model space more thoroughly we show how the use of hierarchical priors provides a flexible and powerful way to focus the search on a reasonable class of models the proposed methodology is demonstrated with four examples three of which come from actual industrial experiments

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

subrepresentation semirings and an analog of 6j symbols let v be a complex representation of the compact group g the subrepresentation semiring associated to v is the set of subrepresentations of the algebra of linear endomorphisms of v with operations induced by the matrix operations the study of these semirings has been motivated by recent advances in materials science in which the search for microstructure independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group so 3 in this case the structure constants for subrepresentation semirings can be described explicitly in terms of the 6j symbols familiar from the quantum theory of angular momentum in this paper we investigate subrepresentation semirings for the class of quasisimply reducible groups defined by mackey multiplicity free representations of finite groups pac j math 8 503 1958 we introduce a new class of symbols called twisted 6j symbols for these groups and we explicitly calculate the structure constan

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

the pivotal cover and frobenius schur indicators abstract in this paper we introduce the notion of the pivotal cover c piv of a left rigid monoidal category c to develop a theoretical foundation for the theory of frobenius schur fs indicators in non pivotal settings for an object v c piv the n r th fs indicator n r v is defined by generalizing that of an object of a pivotal monoidal category this notion gives a categorical viewpoint to some recent results on generalizations of fs indicators based on our framework we also study the fs indicators of the adjoint object in a finite tensor category which can be considered as a generalization of the adjoint representation of a hopf algebra the indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor

some behaviors of fsz groups under central products central quotients and regular wreath products abstract we show that any group g with a non f s z m quotient by a central cyclic subgroup also provides a non f s z m group of order m g obtained as a central product of g with a cyclic group we then construct for every prime p 3 and j n an f s z p j group f such that there is a central cyclic subgroup a with f a not f s z p j we apply these results to regular wreath products to construct an fsz p group which is not f s z for any prime p 3 these give the first known examples of fsz groups that are not f s z we are also able to prove a few partial results concerning the fsz properties for the sylow subgroups of symmetric groups in the appendix we enumerate all non fsz groups of order 57

on the even powers of the antipode of a finite dimensional hopf algebra abstract the trace of powers of the square of the antipode s 2 of a finite dimensional hopf algebra a over a field k is studied it is shown in many cases that the trace function vanishes on s 2 m when s 2 m 1 a finer properties of the antipode are related to this phenomenon

frobenius schur indicators for subgroups and the drinfel d double of weyl groups if g is any finite group and k is a field there is a natural construction of a hopf algebra over k associated to g the drinfel d double d g we prove that if g is any finite real reflection group with drinfel d double d g over an algebraically closed field k of characteristic not 2 then every simple d g module has frobenius schur indicator 1 this generalizes the classical results for modules over the group itself we also prove some new results about weyl groups in particular we prove that any abelian subgroup is inverted by some involution also if e is any elementary abelian 2 subgroup of the weyl group w then all representations of cw e are defined over q

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

a robust criterion for experimental designs for serially correlated observations a new robust criterion for experimental designs is proposed for a given design the criterion is to minimize over all possible run orders the absolute value of the change of the variance function due to possible correlation between the observations the resulting design is robust against possible auto correlation among the observations in the sense that confidence interval coverage levels are maintained accurately applications are given for 2k factorial designs and one mixed level fractional experiment and robust runs are obtained computational strategies are also discussed and a simulated annealing algorithm is developed to search for robust designs

main effect plans orthogonal through the block factor many industrial experiments require different number of levels for different factors orthogonal main effect plans omeps for such asymmetrical experiments often require large run size if blocking is needed then run size usually becomes even larger in this article we show that there are situations when use of blocks may in fact be helpful in finding an omep with small run size explicitly we obtained saturated omeps for a 33n 23n experiment in 3n blocks of size 4 each for each hadamard number n in each of these plans the three level factors are nonorthogonal to the block factor but are pairwise orthogonal through the block factor the two level factors are orthogonal to the block factor the data analysis for such a plan is illustrated with the help of examples this article has supplementary material online

linear stochastic differential equations and wick products we establish the existence and uniqueness of the solution to a multidimensional linear skorohod stochastic differential equation with deterministic diffusion matrix using the notions of wick product andstransform if the diffusion matrix is constant and has real eigenvalues the solution is a stochastic process with moments of all orders provided that the initial condition is differentiable up to a suitable order the case of a diffusion matrix in the first wiener chaos is discussed in the last section

twisted frobenius schur indicators for hopf algebras abstract the classical frobenius schur indicators for finite groups are character sums defined for any representation and any integer m 2 in the familiar case m 2 the frobenius schur indicator partitions the irreducible representations over the complex numbers into real complex and quaternionic representations in recent years several generalizations of these invariants have been introduced bump and ginzburg building on earlier work of mackey have defined versions of these indicators which are twisted by an automorphism of the group in another direction linchenko and montgomery have defined frobenius schur indicators for semisimple hopf algebras in this paper the authors construct twisted frobenius schur indicators for semisimple hopf algebras these include all of the above indicators as special cases and have similar properties

optimal blocking of two level factorial designs summary blocking of two level factorial designs is considered for block sizes 2 and 4 using the method of fractional partial confounding a d and e optimal designs are obtained for block size 2 within the class of orthogonal designs for which main effects and two factor interactions are all orthogonal to each other before allowing for blocking a d and e optimal designs are obtained for block size 4 within the class of orthogonal designs with main effects orthogonal to blocks the designs obtained also have other favourable properties including orthogonal estimation of effects and orthogonality to superblocks

exact relations for effective tensors of composites necessary conditions and sufficient conditions typically the elastic and electrical properties of composite materials are strongly microstructure dependent so it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes they can also be regarded as fundamental invariances existing in a given physical context classic examples include hill s formulae for the effective bulk modulus of a two phase mixture when the phases have equal shear moduli levin s formulae linking the effective thermal expansion coefficient and effective bulk modulus of two phase mixtures and dykhne s result for the effective conductivity of an isotropic two dimensional polycrystalline material here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones the search for exact relations is reduced to a search for matrix subspaces having a structure of special jordan algebras one of many new exact relations is for the effective shear modulus of a class of three dimensional polycrystalline materials we present complete lists of exact relations for three dimensional thermoelectricity and for three dimensional thermopiezoelectric composites that include all exact relations for elasticity thermoelasticity and piezoelectricity as particular cases 2000 john wiley sons inc

nearly orthogonal arrays with mixed levels and small runs in running a factorial experiment it may be desirable to use an orthogonal array with different mixed numbers of factor levels because of the orthogonality requirement such arrays may have a large run size by slightly sacrificing the orthogonality requirement we can obtain nearly orthogonal arrays with economic run size some general methods for constructing such arrays are given for 12 18 20 and 24 runs many orthogonal arrays and nearly orthogonal arrays with mixed levels are constructed and tabulated effects of near orthogonality on estimation efficiency and analysis are studied

columnwise pairwise algorithms with applications to the construction of supersaturated designs motivated by the construction of supersaturated designs we develop a class of algorithms called columnwise pairwise exchange algorithms they differ from the k exchange algorithms in two respects 1 they exchange columns instead of rows of the design matrix and 2 they employ a pairwise adjustment in the search for a better column the proposed algorithms perform very well in the construction of supersaturated designs both for a single criterion and for multiple criteria they are also applicable to the construction of designs that are not supersaturated

k circulant supersaturated designs a class of supersaturated designs called k circulant designs is explored these designs are constructed from cyclic generators by cycling k elements at a time the class of designs includes many es2 optimal designs some of which are already known and some of which are more efficient than known designs for model estimation under factor sparsity generators for the most efficient designs are listed and projection properties of some of the designs are explored we also illustrate that some k circulant supersaturated designs can be augmented with interaction columns to produce efficient designs for a larger number of factors or for estimating interactions

on the existence of saturated and nearly saturated asymmetrical orthogonal arrays we develop a combinatorial condition necessary for the existence of a saturated asymmetrical orthogonal array of strength 2 this condition limits the choice of integral solutions to the system of equations in the bose bush approach and can thus strengthen considerably the bose bush approach as applied to a symmetrical part of such an array as a consequence several nonexistence results follow for saturated and nearly saturated orthogonal arrays of strength 2 one of these leads to a partial settlement of an issue left open in a paper by wu zhang and wang nonexistence of a class of saturated asymmetrical orthogonal arrays of strength 4 is briefly discussed

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

self dual modules of semisimple hopf algebras abstract we prove that over an algebraically closed field of characteristic zero a semisimple hopf algebra that has a nontrivial self dual simple module must have even dimension this generalizes a classical result of w burnside as an application we show under the same assumptions that a semisimple hopf algebra that has a simple module of even dimension must itself have even dimension

an approach to the construction of asymmetrical orthogonal arrays abstract use of asymmetrical orthogonal arrays oa s for planning industrial experiments with mixed levels has become increasingly popular in addition to applications to experimental investigations in many disciplines they can also be used in the balanced repeated replications method of inference from general stratified survey samples in this article we propose a general approach to the construction of asymmetrical oa s with economic run size and flexibility in the choice of factor levels as applications of this approach we construct several general classes of arrays that include numerous existing classes of arrays as special cases a large number of known asymmetrical oa s can be reproduced by our approach in a unified and simple manner many new arrays are obtained a list of asymmetrical oa s with moderate run size is given our approach consists of three steps we first construct an oa as the generalized kronecker sum of an oa and some difference matrices we then add to the constructed oa co

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

the idle column method design construction properties and comparisons taguchi s idle column method for constructing arrays is investigated and generalized a general construction method is proposed via a necessary and sufficient condition on two level orthogonal arrays to which the method is applicable a method is developed for identifying the idle column and pairs of columns in the arrays mixed two and three level idle column arrays with 64 or fewer runs are tabulated some guidelines are offered on when the idle column arrays should be used and when they should be replaced by orthogonal main effect plans or d optimal designs

blocking in regular fractional factorials a projective geometric approach a projective geometric characterization is given of the existence of any regular main effect s n k design in s blocks it leads to a constructive method for finding a maximal blocking scheme for any given fractional factorial design a useful sufficient condition for admissible block designs is given in terms of the minimum aberration property of a certain unblocked design

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

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

hochschild cohomology and the coradical filtration of pointed coalgebras applications abstract in this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the hochschild cohomology of that coalgebra with coefficients in some one dimensional bicomodules as an application for a given prime numberpand an algebraically closed fieldkof characteristic 0 we classify all pointed hopf algebras of dimensionp3overk

stochastic heat equation driven by fractional noise and local time the aim of this paper is to study the d dimensional stochastic heat equation with a multiplicative gaussian noise which is white in space and has the covariance of a fractional brownian motion with hurst parameter h 0 1 in time two types of equations are considered first we consider the equation in the ito skorohod sense and later in the stratonovich sense an explicit chaos expansion for the solution is obtained on the other hand the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the brownian motion

anticipative girsanov transformations the transformations of measures induced by left w int limits 0 bullet k s ds right with ks possibly anticipating the wiener process ws is discussed and a girsanovtype theorem under rather weak assumptions on ks is derived

cosemisimple bialgebras and discrete quantum semigroups abstract we show for a cosemisimple bialgebra that a standard operation making it into a discrete quantum semigroup must be unique it may not exist we prove such an operation on a cosemisimple o q sl2 c exists if and only if the parameterqis real we also conclude that discrete quantum groups form a more restrictive class than cosemisimple hopf algebras

involutory hopf algebras in 1975 kaplansky conjectured that a finite dimensional semisimple hopf algebra is necessarily involutory twelve years later larson and radford proved the conjecture in characterisitic 0 and obtained significant partial results in positive characteristics the goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites no harpoons and gratifyingly few hits let h be a finite dimensional hopf algebra over a field k and let s h h denote its antipode then h is said to be involutory if s2 idh or equivalently if s is an involution on h in k kaplansky conjectured that a semisimple hopf algebra is necessarily involutory and in a fundamental paper lr2 larson and radford proved this assertion at least for fields of characteristic 0 furthermore they showed that if h is semisimple and cosemisimple and if chark p dimk h 2 then h must again be involutory a nice exposition of this material can be found in n while lr2 is reasonably brief it depelids heavily on certain earlier results on characters of hopf algebras as contained in l and lr 1 and the combined proof is substantial the goal of this paper is to present an efficient proof of the main results of lr2 requiring minimal prerequisites as will be apparent our proof follows the same basic outline as that of larson and radford and therefore we offer nothing new in this regard however by computing traces on certain tensor modules we are able to obtain the necessary character formulas quite quickly and this is where the simplifications occur in some sense the hero of this approach is the linear transformation t described below if v is a left h module then h 0 v is well known to be h isomorphic to h 0 vj where vj is equal to v but with the trivial action determined by the counit e this is proved quite simply by considering the map t h v h v given by t h ov hihi h2v for all h e h v e v h indeed one shows that t is invertible with inverse t 1 h x v e hz x s h2 v h received by the editors july 13 1993 and in revised form june 8 1994 originally communicaated to the proceedings of the ams by lance w small 1991 mathematics subject classification primary 1 6w30 research supported in part by nsf grants dms 9224662 and dms 9003374 1995 american mathematical society

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

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

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

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

optimal bounds for conduction in two dimensional multiphase polycrystalline media a complete geometrical characterization is given of the set of all possible effective conductivity tensors for two dimensional composites made of an arbitrary number of given anisotropic phases it is rigorously established that any polycrystalline composite formed from an arbitrary possibly infinite number of phases can be replaced by a composite formed from only two of the given phases without altering the effective conductivity tensor

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

radford s s 4 formula for co frobenius hopf algebras abstract this note extends radford s formula for the fourth power of the antipode of a finite dimensional hopf algebra to co frobenius hopf algebras and studies equivalent conditions to a hopf algebra being involutory for finite dimensional and co frobenius hopf algebras

linear skorohod stochastic differential equations let andb be bounded processes on the wiener space p c 0 1 which are possibly anticipating the brownian motionw t t and let be a bounded random variable we deduce the existence and uniqueness of a solutionx for the linear equation with skorohod integral x t eta int limits 0 t sigma s x s dw s int limits 0 t b s x s ds t in 0 1 1 under rather weak assumptions on and no additional requirement onb and the description of the solutionx requires to study the family t t t 0 1 of transformationt t of into itself associated to 1 by the equation t t omega int limits 0 t wedge sigma s t s omega ds omega in omega t in 0 1

semisimple weak hopf algebras abstract we develop the theory of semisimple weak hopf algebras and obtain analogues of a number of classical results for ordinary semisimple hopf algebras we prove a criterion for semisimplicity and analyze the square of the antipode s 2 of a semisimple weak hopf algebra a we explain how the frobenius perron dimensions of irreducible a modules and eigenvalues of s 2 can be computed using the inclusion matrix associated to a a trace formula of larson and radford is extended to a relation between the categorical and frobenius perron dimensions of a finally an analogue of the class equation of kac and zhu is established and properties of a module algebras and their dimensions are studied

on hopf algebras of dimension 4p abstract in this paper we prove that a non semisimple hopf algebra h of dimension 4 p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided h contains more than two group like elements in particular we prove that non semisimple hopf algebras of dimensions 20 28 and 44 are pointed or their duals are pointed and this completes the classification of hopf algebras in these dimensions

noetherian pi hopf algebras are gorenstein we prove that every noetherian affine pi hopf algebra has finite injective dimension which answers a question of brown 1998

optimal main effect plans with non orthogonal blocking the current literature on fractional factorial plans in block designs centres around orthogonal blocking which may not however always be attainable because of practical restrictions on the block size for general factorials including asymmetric ones sufficient conditions are indicated in this paper for a main effect plan to be universally optimal under possibly non orthogonal blocking a construction procedure is given using generalised youden designs in conjunction with orthogonal arrays we also illustrate how the procedure can be applied to obtain optimal main effect plans in the practically important situation where each factor has two or three levels and the block size is small

infinite families of potential modular data related to quadratic categories we present several infinite families of potential modular data motivated by examples of drinfeld centers of quadratic categories in each case the input is a pair of involutive metric groups with gauss sums differing by a sign along with some conditions on the fixed points of the involutions and the relative sizes of the groups from this input we construct s and t matrices which satisfy the modular relations and whose verlinde coefficients are non negative integers we also check certain restrictions coming from frobenius schur indicators these families generalize evans and gannon s conjectures for the modular data associated to generalized haagerup and near group categories for odd groups and include the modular data of the drinfeld centers of almost all known quadratic categories in addition to the subfamilies conjecturally realized by centers of quadratic categories these families include many examples of potential modular data which do not correspond to known types of modular tensor categories

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

exact relations for effective tensors of polycrystals i necessary conditions the set of all effective moduli of a polycrystal usually has a nonempty interior when it does not we say that there is an exact relation for effective moduli this can indeed happen as evidenced by recent results 4 10 12 on polycrystals in this paper we describe a general method for finding such relations for effective moduli of laminates the method is applicable to any physical setting that can be put into the hilbert space framework developed by milton 13 the idea is to use the w function of milton 13 that transforms a lamination formula into a convex combination the method reduces the problem of finding exact relations to a problem from representation theory of so d d 2 or 3 corresponding to a particular physical setting when this last problem is solved there is a finite amount of calculation required to be done in order to answer the question completely at present each candidate relation has to be examined separately in order to confirm the stability under homogenization we apply our general theory to the settings of conductivity and two dimensional elasticity

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

monoidal morita invariants for finite group algebras abstract two hopf algebras are called monoidally morita equivalent if module categories over them are equivalent as linear monoidal categories we introduce monoidal morita invariants for finite dimensional hopf algebras based on certain braid group representations arising from the drinfeld double construction as an application we show for any integer n the number of elements of order n is a monoidal morita invariant for finite group algebras we also describe relations between our construction and invariants of closed 3 manifolds due to reshetikhin and turaev

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

on the quasitriangular structures of a semisimple hopf algebra in this paper we show that the theorem of 4 on the order of the group of hopf algebra automorphisms aut a of certain semisimple hopf algebras over a field k has an interesting implication for quantum groups if a is a finite dimensional hopf algebra over the field k there may be infinitely many re a a such that a r is a quasitriangular hopf algebra however if a is semisimple and the characteristic of k is 0 or if a is semisimple cosemisimple and involutory and the characteristic of k is p dim a we show in the theorem of this paper that there are at most finitely r ea a such that a r is quasitriangular our proof is based on the fact that the group aut a is finite 4 theorem 33 for such a hopf algebra a we will assume that the reader is familiar with the elementary aspects of the theory of hopf algebras a good general reference is s

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

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

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

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

s4 symmetry of 6j symbols and frobenius schur indicators in rigid monoidal c categories we show that a left rigid monoidal c category with irreducible monoidal unit is also a sovereign and spherical category defining a frobenius schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects as a main result we prove s4 invariance of 6j symbols in such a category