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

testing for the constancy of parameters over time abstract tests are proposed for detecting possible changes in parameters when the observations are obtained sequentially in time while deriving the tests the alternative one has in mind specifies the parameter process as a martingale the distribution theory of these tests relies on the large sample results that is only the limiting null distributions are known except in very special cases the main tool in establishing these limiting distributions is weak convergence of stochastic processes suppose that we have vector valued observations x 1 x n obtained sequentially in time or ordered in some other linear fashion their joint distribution is described by determining the initial distribution for x 1 and the conditional distribution for each x k given the past up to x k 1 suppose further that these distributions depend on a p dimensional parameter vector at least locally i e in a short time period this may be more or less legitimate in the long run however the possibility of some ch

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

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

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

comparisons of tests for the presence of random walk coefficients in a simple linear model abstract the locally most powerful test is derived for the hypothesis that the regression coefficients are constant over time against the alternative that they vary according to the random walk process when the regression equation contains the constant term only comparisons are made with the tests suggested by lamotte and mcwhorter 1978 these are based on exact powers and on three different types of asymptotic efficiencies including the classical pitman and bahadur approaches and the new one due to gregory 1980 the concept of the bahadur efficiency is extended to cover also the random slopes suggestions are made for choosing the test

testing for deterministic trend and seasonal components in time series models summary a univariate time series model can be set up as the sum of trend seasonal and irregular components the trend and seasonal components will normally be stochastic but deterministic components arise as a special case this paper develops a test that the trend and seasonal components are deterministic using the approach of lehmann the procedure is then extended to test for deterministic components in a model formulated in first differences both tests are exact and critical values are tabulated

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

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

rankin selberg without unfolding and bounds for spherical fourier coefficients of maass forms in this paper we study periods of automorphic functions we present a new method which allows one to obtain non trivial spectral identities for weighted sums of certain periods of automorphic functions these identities are modelled on the classical identity of r rankin ra and a selberg se we recall that the rankinselberg identity relates the weighted sum of fourier coefficients of a cusp form to the weighted integral of the inner product of with the eisenstein series e g formula 1 7 below we show how to deduce the classical rankin selberg identity and similar new identities from the uniqueness principle in representation theory also known under the following names the multiplicity one property gelfand pair the uniqueness principle is a powerful tool in representation theory it plays an important role in the theory of automorphic functions we associate a non trivial spectral identity to certain pairs of different triples of gelfand subgroups namely we associate a spectral identity see formula 1 4 below with two triples f h1 g and f h2 g of subgroups in a group g such that pairs g hi and hi f for i 1 2 are strong gelfand pairs having the same subgroup f in the intersection for the notion of gelfand pair that we use see section 1 1 3 we call such a collection g h1 h2 f a strong gelfand formation in the introduction we explain our general idea and describe how to implement it in order to reprove the classical rankin selberg formula we also obtain a new anisotropic analog of the rankin selberg formula we present then an analytical application of these spectral identities towards non trivial bounds for various fourier coefficients of cusp forms the novelty of our results lies mainly in the method as we do not rely on the well known technique of rankin and selberg

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

tests for seasonal moving average unit root in arima models abstract procedures to test for a unit root in the seasonal moving average ma operator for seasonal autoregressive integrated moving average arima models are examined such procedures are useful to distinguish deterministic from stochastic seasonal patterns the simple integrated seasonal ma 1 model z t y t yt s 1 b s t where y t is the observed seasonal series and t is a white noise sequence is considered and tests for h 0 1 versus h 1 1 are studied the locally best invariant unbiased and point optimal invariant tests are investigated the finite sample and asymptotic distributions of the test statistics under both the null hypothesis and local alternatives are derived quantiles of the null distributions are tabulated for quarterly and monthly data cases and a limited study of the powers of the tests is reported the test procedures are extended to more general seasonal arma models of the form z t 1 b s nt where n t follows a general stationary and invertible

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

asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative one component of the coefficients follows a random walk process we discuss the limiting null behavior of the test statistic without assuming normality under two situations where the initial value of the random walk process is known or unknown the limiting distribution is that of a quadratic functional of brownian motion and the characteristic function is obtained from the fredhohm determinant associated with a certain integral equation the limiting distribution is then computed by numerical inversion of the characteristic function

median unbiased estimation of coefficient variance in a time varying parameter model abstract this article considers inference about the variance of coefficients in time varying parameter models with stationary regressors the gaussian maximum likelihood estimator mle has a large point mass at 0 we thus develop asymptotically median unbiased estimators and asymptotically valid confidence intervals by inverting quantile functions of regression based parameter stability test statistics computed under the constant parameter null these estimators have good asymptotic relative efficiencies for small to moderate amounts of parameter variability we apply these results to an unobserved components model of trend growth in postwar u s per capita gross domestic product the mle implies that there has been no change in the trend growth rate whereas the upper range of the median unbiased point estimates imply that the annual trend growth rate has fallen by 0 9 per annum since the 1950s

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

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

optimal tests when a nuisance parameter is present only under the alternative this paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null for example the results apply to tests of structural change with unknown changepoint the testing problem considered is nonstandard and the classical asymptotic optimality results for the lagrange multiplier wald and likelihood ratio do not apply a weighted average power criterion is used here to generate optimal tests this criterion is similar to that used by a wald 1943 to obtain the classical asymptotic optimality properties of wald tests in regular testing problems copyright 1994 by the econometric society

consistent factor estimation in dynamic factor models with structural instability abstract this paper considers the estimation of approximate dynamic factor models when there is temporal instability in the factor loadings we characterize the type and magnitude of instabilities under which the principal components estimator of the factors is consistent and find that these instabilities can be larger than earlier theoretical calculations suggest we also discuss implications of our results for the robustness of regressions based on the estimated factors and of estimates of the number of factors in the presence of parameter instability simulations calibrated to an empirical application indicate that instability in the factor loadings has a limited impact on estimation of the factor space and diffusion index forecasting whereas estimation of the number of factors is more substantially affected

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

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

geodesic restrictions for the casimir operator abstract we consider the hyperbolic casimir operator c defined on the tangent sphere bundle sy of a compact hyperbolic riemann surface y we prove a non trivial bound on the l 2 norm of the restriction of eigenfunctions of c to certain natural hypersurfaces in sy the result that we obtain goes beyond known sharp local bounds of l hormander

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

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

maximum likelihood estimation of regression models with first order moving average errors when the root lies on the unit circle1 this paper considers the maximum likelihood estimator of the first order moving average process when the true value of the coefficient is one the results are also extended to regression analysis it is shown that there is a local maximum of the likelihood function within an interval of o t 1 of the true value and also that the probability that the maximum occurs exactly at the true value can be calculated in finite samples in this paper we consider the maximum likelihood estimator mle of the coefficient p of a first order moving average ma 1 process when the true value of the coefficient is plus or minus one the results are also extended to regression equations with errors generated by the ma 1 process our study has been motivated chiefly by two factors first it has been sometimes suggested that the occurrence of a maximum near or at p 1 could be interpreted as evidence for over differencing 4 13 while this does not present any difficulty for exact maximum likelihood estimation procedures 12 we need to consider the properties of the maximum likelihood estimates of p when true p 1 in order to apply conventional test procedures e g the likelihood ratio test second kang 9 has shown that the likelihood function is stationary at p 1 and simulation studies on the behavior of the mle of p reveal that the global maximum can occur at p 1 even if the true p is within the invertibility region 9 5 there are two main results in this paper first we show that there is a local maximum of the likelihood function p such that p 1 is of o t 1 but p is not asymptotically normally distributed second it is shown possible to calculate the exact probability that p 1 is a local maximum of the likelihood function in finite samples although we focus only on local maxima near or at p 1 by constructing arguments similar to those of jennrich 8 or sargan 14 it can be shown that the global maximum has probability limit one in large samples so that the properties of these maxima would seem to be of primary interest asymptotically in considering the global mle

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

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

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

on a quantum chaos theorem of r schrader and m taylor abstract we extend results of r schrader and m taylor on the semi classical asymptotics of eigenfunctions of a quantum hamiltonian for a particle in a gauge field our method is based on the guillemin sternberg uribe theory of fuzzy ladders and reductions of hamiltonian systems with symmetry

locally best invariant tests of the error covariance matrix of the linear regression model summary this paper considers a class of hypothesis testing problems concerning the covariance matrix of the disturbances in the classical linear regression model a test that is locally best invariant against one sided alternative hypotheses is constructed and shown to be identical to a one sided version of the lagrange multiplier test

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

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

estimation and testing of euler equation models with time varying reduced form coefficients abstract this paper studies the estimation and testing of euler equation models in the framework of the classical two step minimum distance method the time varying reduced form model in the first step reflects the adaptation of private agents beliefs to the changing economic environment the presumed ability of euler conditions to deliver stable parameters indexing tastes and technology is interpreted as a time invariant second step model this paper shows that complementary to and independent of one another both standard specification test and stability test are required for the evaluation of an euler equation as an empirical application a widely used investment euler equation is submitted to examination the empirical outcomes appear to suggest that the standard investment model has not been a success for aggregate investment

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

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

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

laplace beltrami operator on a riemann surface and equidistribution of measures we consider the laplace beltrami operator on a compact riemann surface of a constant negative curvature for any eigenvalue of the laplace beltrami operator there is an associated sequence of measures on the riemann surface these measures naturally appear in quantum chaos type questions in the theory of electro magnetic flow on a riemann surface the main result of the paper is the claim that this sequence of measures has the liouville measure as the weak limit we prove a quantitative version of this equidistribution claim

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

subconvexity bounds for triple l functions and representation theory we describe a new method to estimate the trilinear period on automorphic representations of pgl 2 such a period gives rise to a special value of the triple l function we prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value of the triple l function our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of pgl 2

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure