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

duals of compact lie groups realized in the cuntz algebras and their actions on c algebras abstract as a first step towards a new duality theorem for compact groups we consider a representation category t g of a compact lie group g u d c whose objects are the tensor powers of the defining representation and whose arrows are the intertwiners we associate to t g in a natural way a c algebra og which can be identified with the fixed points of the cuntz algebra od under the natural action of g og carries a canonical endomorphism g and t g can be reconstructed from og g if g su d then og is simple we give conditions for an endomorphism of a unital c algebra a to determine an action of t su d on a such actions correspond to monomorphisms of osu d into a with natural intertwining properties

error calculation of the hosvd based rule base reduction in hierarchical fuzzy systems abstract in real time evaluation systems the decision should be made in time to avoid the serious consequences in the literature there are several techniques which aim to reduce the computational complexity of these kinds of systems among others the higher order singular value decomposition hosvd based reduction method is a useful tool for handling this problem in the case when only non exact reduction can be performed the error calculation is also essential because the reduction degree is based on the calculated error in these systems in this paper the authors present the hosvd reduction error calculation in hierarchical fuzzy systems using mamdani type inference and define a general formula which includes the propagated error from the previous levels of the hierarchy in which the error bound calculated by different ways can be used and takes into account the case when not all the inputs contain a propagated error but only a few of them furthermore a greedy algorithm is presented for reduction optimization

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

on the category o for rational cherednik algebras we study the category o of representations of the rational cherednik algebra aw attached to a complex reflection group w w e construct an exact functor called knizhnik zamolodchikov functor o hw mod where hw is the finite iwahori hecke algebra associated to w we prove that the knizhnik zamolodchikov functor induces an equiv alence between o otor the quotient of o by the subcategory of aw modules supported on the discriminant and the category of finite dimensional hw modules the standard aw modules go under this equivalence to certain modules arising in kazhdan lusztig theory of cells provided w is a weyl group and the hecke algebra hw has equal parameters we prove that the category o is equivalent to the module category over a finite dimensional algebra a generalized q schur algebra associated to w

the c algebra of a vector bundle and fields of cuntz algebras abstract we study the cuntz pimsner algebra associated with the module of continuous sections of a hilbert bundle and prove that it is a continuous bundle of cuntz algebras furthermore we assign to bundles of cuntz algebras carrying a global circle action a class in the representable kk group of the zero grade bundle we explicitly compute such class for the cuntz pimsner algebra of a vector bundle

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

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 ramification of centres lie algebras in positive characteristic and quantised enveloping algebras let h be a hopf algebra over the field k which is a finite module over a central affine sub hopf algebra r examples include enveloping algebras u mathfrak g of finite dimensional k lie algebras mathfrak g in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity the ramification behaviour of the maximal ideals of z h with respect to the subalgebra r is studied and the conclusions are then applied to the cases of classical and quantised enveloping algebras in the case of u mathfrak g for mathfrak g semisimple a conjecture of humphreys 28 on the block structure of u mathfrak g is confirmed in the case of u epsilon mathfrak g for mathfrak g semisimple and epsilon an odd root of unity we obtain a quantum analogue of a result of mirkovic and rumynin 35 and we fully describe the factor algebras lying over the regular sheet 9 the blocks of u epsilon mathfrak g are determined and a necessary condition which may also be sufficient for a baby verma u epsilon mathfrak g module to be simple is obtained

endomorphisms of c algebras cross products and duality for compact groups on definit une nouvelle sorte de produit croise d une c algebre par des semigroupes d endomorphismes avec des proprietes reliees a la symetrie de permutation et a l existence de conjugues un groupe compact est alors defini intrinsequement par son action duale sur le produit croise cette construction donne une caracterisation des duaux de groupes compacts abstraits qui est hors d atteinte de la theorie de dualite de taunaka kein et est independante de cette theorie ceci resout le probleme de demontrer l existence d un groupe de jauge global compact en physique des particules etant seulement donnees les observables locales

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

reflection subgroups of finite complex reflection groups abstract all simple extensions of the reflection subgroups of a finite complex reflection group g are determined up to conjugacy as a consequence it is proved that if the rank of g is n and if g can be generated by n reflections then for every set r of n reflections which generate g every subset of r generates a parabolic subgroup of g

compact lie groups associated with endomorphisms of c algebras on donne des conditions sur un endomorphisme d une c algebre 7b a avec unite et centre trivial en termes d operateurs d entrelacements pour des puissances de cet endomorphisme qui suffisent a determiner un groupe de lie compact g et une action d un g dual sur 7b a

the c0 x algebra of a net and index theory abstract given a connected and locally compact hausdorff space x with a good base k we assign in a functorial way a c 0 x algebra to any precosheaf of c algebras a defined over k afterwards we consider the representation theory and the kasparov k homology of a and interpret them in terms respectively of the representation theory and the k homology of the associated c 0 x algebra when a is an observable net over the spacetime x in the sense of algebraic quantum field theory this yields a geometric description of the recently discovered representations affected by the topology of x

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

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

a recursive rule base adjustment algorithm for a fuzzy logic controller this paper introduces a recursive rule base adjustment to enhance the performance of fuzzy logic controllers here the fuzzy controller is constructed on the basis of a decision table dt relying on membership functions and fuzzy rules that incorporate heuristic knowledge and operator experience if the controller performance is not satisfactory it has previously been suggested that the rule base be altered by combined tuning of membership functions and controller scaling factors the alternative approach proposed here entails alteration of the fuzzy rule base the recursive rule base adjustment algorithm proposed in this paper has the benefit that it is computationally more efficient for the generation of a dt and advantage for online realization simulation results are presented to support this thesis

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

morita homotopy theory of c categories abstract in this article we establish the foundations of the morita homotopy theory of c categories concretely we construct a cofibrantly generated simplicial symmetric monoidal quillen model structure denoted by m mor on the category c 1 cat of small unital c categories the weak equivalences are the morita equivalences and the cofibrations are the functors which are injective on objects as an application we obtain an elegant description of brown green rieffel s picard group in the associated homotopy category ho m mor we then prove that ho m mor is semi additive by group completing the induced abelian monoid structure at each hom set we obtain an additive category ho m mor 1 and a composite functor c 1 cat ho m mor ho m mor 1 which is characterized by two simple properties inversion of morita equivalences and preservation of all finite products finally we prove that the classical grothendieck group functor becomes co represented in ho m mor 1 by the tensor unit object

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

fuzzy logic controller based on genetic algorithms abstract based on genetic algorithms gas a method of designing a fuzzy logical controller for complex processes is proposed after a brief overview of ga the selection of ga s control parameters is discussed for designing mimo fuzzy control systems we explained the optimization procedure of the fuzzy rules selection on a large fuzzy input search space a double inverted pendulum device is used to test the effectiveness of the design method based on gas not only simulation results are shown in the paper but also the practical operation results are given

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

from path representations to global morphisms for a class of minimal models we construct global observable algebras and global dhr morphisms for the virasoro minimal models with central charge c 2 q q odd to this end we pass from the irreducible highest weight modules to path representations which involve fusion graphs of the c 2 q models the paths have an interpretation in terms of quasi particles which capture some structure of non conformal perturbations of the c 2 q models the path algebras associated to the path spaces serve as algebras of bounded observables global morphisms which implement the superselection sectors are constructed using quantum symmetries we argue that there is a canonical semi simple quantum symmetry algebra for each quasi rational cft in particular for the c 2 q models these symmetry algebras act naturally on the path spaces which allows to define a global field algebra and covariant multiplets therein

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

blocks of restricted rational cherednik algebras for g m d n abstract we study the dunkl opdam subalgebra of the rational cherednik algebra for wreath products at t 0 and use this to describe the block decomposition of restricted rational cherednik algebras for g m d n

cuspidal representations of rational cherednik algebras at t 0 we study those finite dimensional quotients of the rational cherednik algebra at t 0 that are supported at a point of the centre it is shown that each such quotient is morita equivalent to a certain cuspidal quotient of a rational cherednik algebra associated to a parabolic subgroup of w

brauer type reciprocity for a class of graded associative algebras classical brauer reciprocity can be stated roughly as follows let g be a finite group and let k be an algebraically closed field of characteristic p 0 if s is a simple kg module and p s is its projective cover then the multiplicity of a simple module l as a composition factor of a characteristic zero lift of p s is the same as the multiplicity of s as a composition factor of a modulo p reduction of l all modules in this paper are assumed to be finite dimensional one thinks of the module l as playing an intermediate role between p s and s reciprocities similar to brauer reciprocity here called brauer type reciprocities have subsequently been found to occur in many other settings for instance if g is a classical modular lie algebra definition given in section 2 then there exists a set 9 of g modules with the following property given a simple g module s the projective cover p s of s has a filtration with each successive quotient isomorphic to a module in 5 and for each such filtration the number of times ze 9 occurs is the same as the multiplicity of s as a composition factor of z this was proved for g of type a i by pollack lo and for arbitrary g by humphreys 3 the reciprocity in this setting is often called humphreys reciprocity inspired by humphreys result bernstein gelfand and gelfand sought and found a brauer type reciprocity bgg reciprocity in a certain truncated category their category 8 of modules for a complex semisimple lie algebra see 11 later this was generalized by mirollo and

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

genetic synthesis of fuzzy logic controllers in turning abstract the paper presents an optimal fuzzy logic controller design using efficient robust optimization techniques called genetic algorithms it is shown that genetic algorithms can automatically search input and output scaling factors membership functions and fuzzy rules of the fuzzy logic controllers based on a fitness function as a result fuzzy logic controllers with optimal control performance can be systematically constructed instead of using a time consuming trial and error approach an adaptive force control system in turning operations is then used to illustrate the proposed method it is shown that the developed fuzzy logic controller can achieve an automatic adjustment of feed rate to optimize the production rate with a constant cutting force in turning operations

the fusion rules of some free wreath product quantum groups and applications abstract in this article we find the fusion rules of the free wreath product quantum groups s n for any discrete group to do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations we then apply the knowledge of the fusion rules to prove in most cases several operator algebraic properties of the associated reduced c algebras such as simplicity and uniqueness of the trace we also prove that the associated von neumann algebra is a full type ii 1 factor and that the dual of s n has the haagerup approximation property for all finite groups

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

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

simultaneous design of membership functions and rule sets for fuzzy controllers using genetic algorithms this paper examines the applicability of genetic algorithms ga s in the simultaneous design of membership functions and rule sets for fuzzy logic controllers previous work using genetic algorithms has focused on the development of rule sets or high performance membership functions however the interdependence between these two components suggests a simultaneous design procedure would be a more appropriate methodology when ga s have been used to develop both it has been done serially e g design the membership functions and then use them in the design of the rule set this however means that the membership functions were optimized for the initial rule set and not the rule set designed subsequently ga s are fully capable of creating complete fuzzy controllers given the equations of motion of the system eliminating the need for human input in the design loop this new method has been applied to two problems a cart controller and a truck controller beyond the development of these controllers we also examine the design of a robust controller for the cart problem and its ability to overcome faulty rules

cuspidal calogero moser and lusztig families for coxeter groups the goal of this paper is to compute the cuspidal calogero moser families for all infinite families of finite coxeter groups at all parameters we do this by first computing the symplectic leaves of the associated calogero moser space and then by classifying certain rigid modules numerical evidence suggests that there is a very close relationship between calogero moser families and lusztig families our classification shows that additionally the cuspidal calogero moser families equal cuspidal lusztig families for the infinite families of coxeter groups

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

orderings on calogero moser partition of imprimitive groups we extend to all parameters the constructions of the geometric and combinatorial orders on irrg l 1 n due to 14 as well as the relations with the a and c functions this allows us to generalize these properties for the group g l e n at least for e n

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

highest weight theory for finite dimensional graded algebras with triangular decomposition abstract we show that the category of graded modules over a finite dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category when the algebra is self injective we show furthermore that this highest weight category has tilting modules in the sense of ringel this provides a new perspective on the representation theory of such algebras and leads to several new structures attached to them there are a wide variety of examples in algebraic lie theory to which this applies restricted enveloping algebras lusztig s small quantum groups hyperalgebras finite quantum groups and restricted rational cherednik algebras

continuous fields of c algebras arising from extensions of tensor c categories abstract the notion of extension of a given c category c by a c algebra a is introduced in the commutative case a c the objects of the extension category are interpreted as fiber bundles over of objects belonging to the initial category it is shown that the doplicher roberts algebra dr algebra in the following associated to an object in the extension of a strict tensor c category is a continuous field of dr algebras coming from the initial one in the case of the category of the hermitian vector bundles over the general result implies that the dr algebra of a vector bundle is a continuous field of cuntz algebras some applications to pimsner c algebras are given

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

reduction of fuzzy control rules by means of premise learning method and case study rule number reduction is important for fuzzy control of complex processes with high dimensionality it is stated in the paper that this issue can be treated effectively by means of learning premises with general structure since conditions of rules are generalised by a genetic algorithm ga rather than enumerated according to every and connection of input fuzzy sets a parsimonious knowledge base with a reduced number of rules can be expected on the other hand to give a numerical evaluation of possible conflicts among rules a consistency index of the rule set is established this index is integrated into the fitness function of the ga to search for a set of optimal rule premises yielding not only good control performance but also little or no inconsistency in the fuzzy knowledge base the advantage of the proposed method is demonstrated by the case study of development of a compact fuzzy controller to balance an inverted pendulum in the laboratory

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

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

genetic fuzzy logic controller an iterative evolution algorithm with new encoding method logic rules and membership functions are two key components of a fuzzy logic controller flc if only one component is learned the other one is often set subjectively thus can reduce the applicability of flc if both components are learned simultaneously a very long chromosome is often needed thus may deteriorate the learning performance to avoid these shortcomings this paper employs genetic algorithms to learn both logic rules and membership functions sequentially we propose a bi level iterative evolution algorithm in selecting the logic rules and tuning the membership functions for a genetic fuzzy logic controller gflc the upper level is to solve the composition of logic rules using the membership functions tuned by the lower level the lower level is to determine the shape of membership functions using the logic rules learned from the upper level we also propose a new encoding method for tuning the membership functions to overcome the problem of too many constraints our proposed gflc model is compared with other similar gflc artificial neural network and fuzzy neural network models which are trained and validated by the same examples with theoretical and field observed car following behaviors the results reveal that our proposed gflc has outperformed