decomposition algebras and axial algebras abstract we introduce decomposition algebras as a natural generalization of axial algebras majorana algebras and the griess algebra they remedy three limitations of axial algebras 1 they separate fusion laws from specific values in a field thereby allowing repetition of eigenvalues 2 they allow for decompositions that do not arise from multiplication by idempotents 3 they admit a natural notion of homomorphisms making them into a nice category we exploit these facts to strengthen the connection between axial algebras and groups in particular we provide a definition of a universal miyamoto group which makes this connection functorial under some mild assumptions we illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given permutation group this construction leads to a large number of examples we also take the opportunity to fix some terminology in this rapidly expanding subject

an analogue of the unitary displacement operator for the q oscillator using the concept of a non standard hilbert space over the quantum complex plane recently introduced by kowalski et al 1993 we construct a unitary q analogue of the weyl displacement operator we investigate the q displaced vacuum states and show that they exhibit properties analogous to coherent states in the undeformed theory they are however distinct from any q coherent states previously found in the literature

quantum algebra deforming maps clebsch gordan coefficients coproducts r and u matrices quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts map induced coproducts are connected to the usual ones by similarity transformations u that may be expressed either in terms of clebsch gordan coefficients or in a universal operator form the product of two such u matrices yields the r matrix for a fixed value of the spectral parameter which bears on the yang baxterization of u as well as r all this is explicitly illustrated for the tensor product 1 2 j of su 2 q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two parameter manifold some observations are made on the general structure of u and r matrices and of coproduct manifolds based on the solutions of hierarchies of partial difference equations applications of deforming maps and u matrices to the physics of spin chains are outlined

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 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

on the grothendieck rings of equivariant fusion categories in this paper we describe a mackey type decomposition for group actions on abelian categories this allows us to define new mackey functors which associates to any subgroup the k theory of the corresponding equivariantized abelian category in the case of an action by tensor autoequivalences the mackey functor at the level of grothendieck rings has a green functor structure as an application we give a description of the grothendieck rings of equivariantized fusion categories under group actions by tensor autoequivalences on graded fusion categories in this settings a new formula for the tensor product of any two simple objects of an equivariantized fusion category is given simplifying the fusion formula from burciu and natale j math phys 54 013511 2013

suq 3 corepresentations and bivariate q krawtchouk polynomials the matrix elements of unitary suq 3 corepresentations which are analogs of the symmetric powers of the natural representation are shown to be the bivariate q krawtchouk orthogonal polynomials thus providing an algebraic interpretation of these polynomials in terms of quantum groups the matrix elements of unitary suq 3 corepresentations which are analogs of the symmetric powers of the natural representation are shown to be the bivariate q krawtchouk orthogonal polynomials thus providing an algebraic interpretation of these polynomials in terms of quantum groups

tensor products of q superalgebra representations and q series identities we work out examples of tensor products for distinct q generalizations of euclidean oscillator and type superalgebras in cases where the method of highest weight vectors will not apply in particular we use the three term recurrence relations for askey wilson polynomials to decompose the tensor product of representations from the positive discrete series and representations from the negative discrete series we show that various q analogues of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and we compute the corresponding matrix elements of the group operators on these representation spaces we show that the matrix elements themselves transform irreducibly under the action of the quantum superalgebra the most important q series identities derived here are interpreted as the expansion of the matrix elements of a group operator via the exponential mapping in a tensor product basis in terms of the matrix elements in a reduced basis they involve q hypergeometric series with base

semisimple hopf algebras of dimension 60 abstract we determine the isomorphism classes of semisimple hopf algebras of dimension 60 which are simple as hopf algebras

the lazy homology of a hopf algebra to any hopf algebra h we associate two commutative hopf algebras h1l h and h2l h which we call the lazy homology hopf algebras of h these hopf algebras are built in such a way that they are predual to the lazy cohomology groups based on the so called lazy cocycles we establish two universal coefficient theorems relating the lazy cohomology to the lazy homology and we compute the lazy homology of the sweedler algebra

on the khintchine constant we present rapidly converging series for the khintchine constant and for general khintchine means of continued fractions we show that each of these constants can be cast in terms of an efficient free parameter series each series involving values of the riemann zeta function rationals and logarithms of rationals we provide an alternative polylogarithm series for the khintchine constant and indicate means to accelerate such series we discuss properties of some explicit continued fractions constructing specific fractions that have limiting geometric mean equal to the khintchine constant we report numerical evaluations of such special numbers and of various khintchine means in particular we used an optimized series and a collection of fast algorithms to evaluate the khintchine constant to more than 7000 decimal places k 1

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

q rotations and other q transformations as unitary nonlinear automorphisms of quantum algebras for the quantum algebras with three generators suq 2 suq 1 1 q oscillator and others the unitary operator u s q is found in terms of q exponents from algebra s generators being the q analog of the linear automorphism for the lie algebras the operator u s q is shown to generate nonlinear automorphisms of the quantum algebras q rotations q shifts bogoliubov q transformations etc the matrix elements tnn of the operator u s q are explicitly expressed in terms of askey wilson polynomials the operator u s q is shown to give rise the new systems of q coherent states of the quantum algebras

the khintchine constants for generalized continued fractions let t p x 1 x p mod 1 for 0p 0 0 241485 then there exists an ergodic invariant measure of the form r pdx let a email protected 1 t p n 1 x email protected n 1 where email protected is the integer part of t if p 1 then a 1 a 2 a n are the partial quotients of the classical continued fraction of x for a real number q we consider averages of a n k p q n x a 1a 2 a n 1 nifq 0 a 1 q a 2 q a n q n 1 qifq 0 we show that i for almost every x k p q lim n k p q n x logk p q p 1 if q 0 where log denotes the natural logarithm iii lim p logk p q logp 1 q if q 0 the limiting behavior of k p q is investigated as email protected 0 with computer simulations

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

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

simple compact quantum groups i the notion of simple compact quantum group is introduced as non trivial noncommutative and noncocommutative examples the following families of compact quantum groups are shown to be simple a the universal quantum groups bu q for q gl n c satisfying qq in n 2 b the quantum automorphism groups aaut b of finite dimensional c algebras b endowed with the canonical trace when dim b 4 including the quantum permutation groups aaut xn on n points n 4 c the standard deformations kq of simple compact lie groups k and their twists kqu as well as rieffel s deformation kj

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

classification of rank 5 premodular categories we survey a number of classification tools developed in recent years and employ them to classify pseudo unitary rank 5 premodular categories up to grothendieck equivalence

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

equivalent notions of normal quantum subgroups compact quantum groups with properties f and fd and other applications abstract the notion of normal quantum subgroup introduced in algebraic context by parshall and wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author as applications a quantum analog of the third fundamental isomorphism theorem for groups is obtained which is used along with the equivalence theorem to obtain results on structure of quantum groups with property f and quantum groups with property fd other results on normal quantum subgroups for tensor products free products and crossed products are also proved

on gauging symmetry of modular categories topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular tensor category umc a group symmetry of the topological phase induces a group symmetry of its corresponding umc gauging is a well known theoretical tool to promote a global symmetry to a local gauge symmetry we give a mathematical formulation of gauging in terms of higher category formalism roughly given a umc with a symmetry group g gauging is a 2 step process first extend the umc to a g crossed braided fusion category and then take the equivariantization of the resulting category gauging can tell whether or not two enriched topological phases of matter are different and also provides a way to construct new umcs out of old ones we derive a formula for the h 4 obstruction prove some properties of gauging and carry out gauging for two concrete examples

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

decomposition of products in nilpotent table algebras abstract two theorems on the decomposition of products in nilpotent commutative table algebras are established this generalizes several results of adan bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p groups and extends these results to fusion rule algebras and association schemes

on the complexity of real root isolation using continued fractions we present algorithmic complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers one motivation is to explain the method s good performance in practice we derive an expected complexity bound of email protected b d 6 d email protected 2 where d is the polynomial degree and t bounds the coefficient bit size using a standard bound on the expected bit size of the integers in the continued fraction expansion thus matching the current worst case complexity bound for real root isolation by exact methods sturm descartes and bernstein subdivision moreover using a homothetic transformation we improve the expected complexity bound to email protected b d email protected we compute the multiplicities within the same complexity and extend the algorithm to non square free polynomials finally we present an open source c implementation in the algebraic library synaps and illustrate its completeness and efficiency as compared to some other available software for this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000

on thin residues and basis digraphs of nilpotent table algebras and applications to nilpotent groups abstract in this paper we study characterizations of nilpotent table algebras in terms of thin residues thin residue matrices thin residue digraphs and basis digraphs we will also discuss applications to nilpotent groups in particular we give a conceptual explanation of the results in takegahara and yoshida 2008 ty

coherent states for transparent potentials the darboux transformation operator method is applied to the investigation of coherent states of transparent multisoliton potentials an isometric correspondence between hilbert spaces of the states of a free particle and a particle moving in the soliton potential is established it is shown that the darboux transformation operator being unbounded and closed cannot realize an isometric mapping between hilbert spaces a quasispectral representation of transformation operators in terms of continuous basis sets is obtained different families of coherent states of the multisoliton potential are introduced measures that realize the resolution of the unity in terms of the projectors on the coherent states vectors are calculated these measures are defined by ordinary smooth functions for the states obtained with the help of bounded transformation operators and by generalized ones otherwise

on the grothendieck rings of generalized drinfeld doubles abstract in this paper it is shown that any irreducible representation of a drinfeld double d a of a semisimple hopf algebra a can be obtained as an induced representation from a certain hopf subalgebra of d a this generalizes a well known result concerning the irreducible representations of drinfeld doubles of finite groups 11 using this description we also give a formula for the fusion rules of semisimple drinfeld doubles this shows that the grothendieck rings of these drinfeld doubles have a ring structure similar to the grothendieck rings of drinfeld doubles of finite groups

computation and theory of extended mordell tornheim witten sums we consider some fundamental generalized mordell tornheim witten mtw zeta function values along with their derivatives and explore connections with multiple zeta values mzvs to achieve this we make use of symbolic integration high precision numerical integration and some interesting combinatorics and special function theory our original motivation was to represent unresolved constructs such as eulerian log gamma integrals we are able to resolve all such integrals in terms of a mtw basis we also present for a substantial subset of mtw values explicit closed form expressions in the process we signicantly extend methods for high precision numerical computation of polylogarithms and their derivatives with respect to order

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

nonlinear shifts of q bose operators and q coherent states the analogue of shift for q bose operators is found in contrast with the case of ordinary bose operators this shift is a nonlinear transformation unitary operators of shift do not form a group two dual systems of q coherent states associated with these shift operators are obtained the matrix elements of the shift operator in canonical basis are expressed via the charlier q polynomials

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

fermionic modular categories and the 16 fold way we study spin and super modular categories systematically as inspired by fermionic topological phases of matter which are always fermion parity enriched and modelled by spin topological quantum field theories at low energy we formulate a 16 fold way conjecture for the minimal modular extensions of super modular categories to spin modular categories which is a categorical formulation of gauging the fermion parity we investigate general properties of super modular categories such as fermions in twisted drinfeld doubles verlinde formulas for naive quotients and explicit extensions of psu 2 4m 2 with an eye towards a classification of the low rank cases

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

more on the q oscillator algebra and q orthogonal polynomials properties of certain q orthogonal polynomials are connected to the q oscillator algebra the wall and q laguerre polynomials are shown to arise as matrix elements of q exponentials of the generators in a representation of this algebra a realization is presented where the continuous q hermite polynomials form a basis of the representation space various identities are interpreted within this model in particular the connection formula between the continuous big q hermite and the continuous q hermite polynomials is thus obtained and two generating functions for these last polynomials are derived algebraically

some properties of group theoretical categories abstract we first show that every group theoretical category is graded by a certain double coset ring as a consequence we obtain a necessary and sufficient condition for a group theoretical category to be nilpotent we then give an explicit description of the simple objects in a group theoretical category following o2 and of the group of invertible objects of a group theoretical category in group theoretical terms finally under certain restrictive conditions we describe the universal grading group of a group theoretical category

irregular primes to one million using fast algorithms for power series inversion based on the fast fourier transform and multisectioning of power series we have calculated all irregular primes up to one million including their indices of irregularity and associated irregular pairs using this data we verified that fermat s last theorem and vandiver s conjecture are true for these primes two primes with index of irregularity five were already known we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six a pair of integers p k is said to be an irregular pair if p is a prime k is an even integer satisfying 2 p are known see 8 our calculations show that vandiver s conjecture is true for all p 106 the table of irregular pairs could also be used to calculate iwasawa invariants for the corresponding primes we did not do this as part of our calculations see 9 or 4 for a discussion of this problem previous computations of irregular pairs have used algorithms that take 0 p2 arithmetic operations for each prime p the bernoulli numbers are defined in received by the editor april 30 1991 and in revised form september 10 1991 1991 mathematics subject classification primary 11b68 secondary 1 1d41 65t20 1 1y99 the first author was partially supported by national science foundation grant dms 9012989 d 1992 american mathematical society 0025 5718 92 1 00 25 per page

free unitary groups are almost simple we show that the quotients of wang and van daele s universal quantum groups by their centers are simple in the sense that they have no normal quantum subgroups thus providing the first examples of simple compact quantum groups with non commutative fusion rings

fast evaluation of multiple zeta sums we show that the multiple zeta sum s 1 s 2 s d n1 n2 nd 1 n s1 1 n s2 2 n sd d for positive integers s i with s 1 1 can always be written as a finite sum of products of rapidly convergent series perhaps surprisingly one may develop fast summation algorithms of such efficiency that the overall complexity can be brought down essentially to that of one dimensional summation in particular for any dimension d one may resolve d good digits of in o d log d log log d arithmetic operations with the implied big o constant depending only on the set s 1 s d

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