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 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

spectral analysis in linear process identification a new look at an old problem the problem of estimating the transfer function frequency response of a linear process by means of a modified spectral analysis method is treated a method of determining the maximum time lags of the correlation functions which makes use of the available a priori physical knowledge about the process and the input signal is proposed this leads in a natural way to the use of an asymmetrical window technique some simulation results are used to illustrate the method based on the asymptotic properties of the estimate the input design is studied finally it will be shown that this work can be used for the estimation of an upper bound of the model uncertainty in the frequency domain

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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