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

categorical formulation of finite dimensional quantum algebras we describe how frobenius monoids give the correct categorical description of certain kinds of finite dimensional quantum algebras we develop the concept of an involution monoid and use it to construct a correspondence between finite dimensional c algebras and certain types of frobenius monoids in the category of hilbert spaces using this technology we recast the spectral theorems for commutative c algebras and for normal operators into an explicitly categorical language and we examine the case that the results of measurements do not form finite sets but rather objects in a finite boolean topos we describe the relevance of these results for topological quantum field theory

extensions of automorphisms and gauge symmetries we characterize the automorphisms of ac algebra open image in new window which extend to automorphisms of the crossed product open image in new window by a compact group dual the case where the inclusion open image in new window is equipped with a group of automorphisms commuting with the dual action is also treated these results are applied to the analysis of broken gauge symmetries in quantum field theory to draw conclusions on the structure of the degenerate vacua on the field algebra

superselection in the presence of constraints superselection and constraints occur together in many gauge theories and here we begin a study of such systems our main focus will be to analyze compatibility questions between constraining and superselection and we will develop an example modelled on qed in which our framework is realized we proceed from a generalization of doplicher roberts superselection theory to the case of the nontrivial center and a set of dirac quantum constraints and find conditions under which the superselection structures will survive constraining in some form this involves an analysis of the restriction and factorization of superselection structures

amenable groups and generalized cuntz algebras withha hilbert space ohthe generalized cuntz algebra overhendowed with the canonical action ofu h we consider a natural family s0of product representations of the zero grade part o0h and the family s obtained from s0by inducing to ohvia the natural conditional expectationp oh o0h each element of s0 and hence of s is labelled by a sequence of unit vectors inh letgbe a non compact locally compact group its regular representation onh l2 g and ohendowed with the action ofg we prove thatgis amenable iff there are ireducible covariant representations in s

ergodic actions of compact groups on operator algebras i general theory we introduce some general machinery for studying ergodic actions of compact groups on operator algebras the most important tool being that of the multiplicity map this theory is used subsequently in 42 to prove that a von neumann algebra on which su 2 acts ergodically is necessarily of type i

compact ergodic groups of automorphisms it is shown that if g is a compact ergodic group of automorphisms on a unital c algebra a then the unique g invariant state is a trace hence if a is a von neumann algebra then it is finite and infective

quantum automorphism groups of homogeneous graphs associated to a finite graph x is its quantum automorphism group g the main problem is to compute the poincare series of g meaning the series f z 1 c1z c2z2 whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation in this paper we find a duality between certain quantum groups and planar algebras which leads to a planar algebra formulation of the problem together with some other results this gives f for all homogeneous graphs having 8 vertices or less

nica toeplitz algebras associated with product systems over right lcm semigroups abstract we prove uniqueness of representations of nica toeplitz algebras associated to product systems of c correspondences over right lcm semigroups by applying our previous abstract uniqueness results developed for c precategories our results provide an interpretation of conditions identified in work of fowler and fowler raeburn and apply also to their crossed product twisted by a product system in the new context of right lcm semigroups as well as to a new doplicher roberts type c algebra associated to the nica toeplitz algebra as a derived construction we develop nica toeplitz crossed products by actions with completely positive maps this provides a unified framework for nica toeplitz semigroup crossed products by endomorphisms and by transfer operators we illustrate these two classes of examples with semigroup c algebras of right and left semidirect products

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

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

the free wreath product of a compact quantum group by a quantum automorphism group in this thesis we define and study the free wreath product of a compact quantum group by a quantum automorphism group and in this way we generalize the previous notion of free wreath product by the quantum symmetric group introduced by bichon our investigation is divided into two part in the first we define the free wreath product of a discrete group by a quantum automorphism group we show how to describe its intertwiners by making use of decorated noncrossing partitions and from this thanks to a result of lemeux we deduce the irreducible representations and the fusion rules then we prove some properties of the operator algebras associated to this compact quantum group such as the simplicity of the reduced c algebra and the haagerup property of the von neumann algebra the second part is a generalization of the first one we start by defining the notion of free wreath product of a compact quantum group by a quantum automorphism group we generalize the description of the spaces of the intertwiners obtained in the discrete case and by adapting a monoidal equivalence result of lemeux and tarrago we find the irreducible representations and the fusion rules then we prove some stability properties of the free wreath product operation in particular we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings we also establish some analytic and algebraic properties of the dual quantum group and of the operator algebras associated to a free wreath product as a last result we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group

simple and prime crossed products of c algebras by compact quantum group coactions let g a be a compact quantum group and a coaction of g on a c algebra b we give necessary and sufficient conditions for the simplicity and primeness of the crossed product b g in terms of certain fixed point algebras

simulation of stochastic reaction diffusion processes on unstructured meshes we model stochastic chemical systems with diffusion by a reaction diffusion master equation on a macroscopic level the governing equation is a reaction diffusion equation for the averages of the chemical species on a mesoscopic level the master equation for a well stirred chemical system is combined with a discretized brownian motion in space to obtain the reaction diffusion master equation the space is covered in our method by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the laplace operator on the macroscale the resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso or on the macroscale level the accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology

on unitary representation theories of compact quantum groups categorical structure of unitary representation of compact quantum groups is studied with relation to a metrical structure encountered in the monoidal category of bimodules of finite jones index

computing the moments of high dimensional solutions of the master equation derived from the markov character only the master equation of chemical reactions is an accurate stochastic description of quite general systems in chemistry exact solutions of this equation are rare and the most frequently used approximative solution method is to write down the corresponding set of reaction rate equations in many cases this approximation is not valid or only partially so as stochastic effects caused by the natural noise present in the full description of the problem are poorly captured in this paper it is shown how a certain set of higher order equations can be derived it is shown by theory and example that stochastic effects are better captured using this technique while still maintaining the computational advantages of the reaction rate approach

a characterization of right coideals of quotient type and its application to classification of poisson boundaries let mathbb g be a co amenable compact quantum group we show that a right coideal of mathbb g is of quotient type if and only if it is the range of a conditional expectation preserving the haar state and is globally invariant under the left action of the dual discrete quantum group we apply this result to the theory of poisson boundaries introduced by izumi for discrete quantum groups and generalize a work of izumi neshveyev tuset on suq n for co amenable compact quantum groups with the commutative fusion rules more precisely we prove that the poisson integral is an isomorphism between the poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of kac type in particular the poisson boundary and the quantum flag manifold are isomorphic for any q deformed classical compact lie group

a solver for the stochastic master equation applied to gene regulatory networks an important driver of gene regulatory networks is noise arising from the stochastic nature of interactions of genes their products and regulators thus such systems are stochastic and can be modelled by the chemical master equations a major challenge is the curse of dimensionality which occurs when one attempts to integrate these equations while stochastic simulation techniques effectively address the curse many repeated simulations are required to provide precise information about stationary points bifurcation phenomena and other properties of the stochastic processes an alternative way to address the curse of dimensionality is provided by sparse grid approximations the sparse grid methodology is applied and the application demonstrated to work efficiently for up to 10 proteins as sparse grid methods have been developed for the approximation of smooth functions a variant for infinite sequences had to be developed together with a multiresolution analysis similar to haar wavelets error bounds are provided which confirm the effectiveness of sparse grid approximations for smooth high dimensional probability distributions

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 uniqueness of limit cycle for the equation ẍ f x φ ẋ ẋ g x 0 abstract in this paper we use the theory of generalized rotated fields to prove a theorem of the uniqueness of limit cycle for the generalized lienard system the damping terms of this system are dependent on x our result extends a famous zhang zhifen s theorem on the uniqueness problem of limit cycle which is contributed the lienard systems whose damping terms are independent of x

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

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

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

superselection sectors and general covariance i this paper is devoted to the analysis of charged superselection sectors in the framework of the locally covariant quantum field theories we shall analyze sharply localizable charges and use net cohomology of j e roberts as a main tool we show that to any 4 dimensional globally hyperbolic spacetime a unique up to equivalence symmetric tensor mathrm c category with conjugates in case of finite statistics is attached to any embedding between different spacetimes the corresponding categories can be embedded contravariantly in such a way that all the charged quantum numbers of sectors are preserved this entails that to any spacetime is associated a unique gauge group up to isomorphisms and that to any embedding between two spacetimes there corresponds a group morphism between the related gauge groups this form of covariance between sectors also brings to light the issue whether local and global sectors are the same we conjecture this holds that at least on simply connected spacetimes it is argued that the possible failure might be related to the presence of topological charges our analysis seems to describe theories which have a well defined short distance asymptotic behaviour

detailed balance as a quantum group symmetry of kraus operators a unital completely positive map governing the time evolution of a quantum system is usually called a quantum channel and it can be represented by a tuple of operators which are then referred to as the kraus operators of the channel we look at states of the system whose correlations with respect to the channel have a certain symmetry then we show that detailed balance holds if the kraus operators satisfy a very interesting algebraic relation which plays an important role in the representation theory of any compact quantum group

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

bundles of c categories we introduce the notions of multiplier c category and continuous bundle of c categories as the categorical analogues of the corresponding c algebraic notions every symmetric tensor c category with conjugates is a continuous bundle of c categories with base space the spectrum of the c algebra associated with the identity object we classify tensor c categories with fibre the dual of a compact lie group in terms of suitable principal bundles this also provides a classification for certain c algebra bundles with fibres fixed point algebras of od

ergodic actions of compact groups on operator algebras iii classification for su 2 we prove that the only ergodic actions ofsu 2 are on type i von neumann algebras using the theory developed in 8

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

quantum group of orientation preserving riemannian isometries we formulate a quantum group analogue of the group of orientation preserving riemannian isometries of a compact riemannian spin manifold more generally of a possibly r twisted and of compact type spectral triple the main advantage of this formulation which is directly in terms of the dirac operator is that it does not need the existence of any good laplacian as in our previous works on quantum isometry groups several interesting examples including those coming from rieffel type deformation as well as the equivariant spectral triples on su 2 and s c2 are discussed

connected components of compact matrix quantum groups and finiteness conditions abstract we introduce the notion of identity component of a compact quantum group and that of total disconnectedness as a drawback of the generalized burnside problem we note that totally disconnected compact matrix quantum groups may fail to be profinite we consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups these sequences have lengths 1 in the classical case but can be countably infinite for duals of discrete groups we give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1 we give necessary and sufficient conditions for normality of the identity component in the sense of wang and finiteness or profiniteness of the quantum component group among them we introduce an ascending chain condition on the representation ring called lie property which characterizes lie groups in the classical case and reduces to group noetherianity of the discrete group dual in the cocommutative case it is weaker than ring noetherianity but ensures existence of a generating representation the lie property and ring noetherianity are inherited by quotient quantum groups we show that a u f is not of lie type we discuss an example arising from the compact real form of u q sl 2 for q 0

some remarks on group bundles and c dynamical systems we introduce the notion of fibred action of a group bundle on a c x algebra by using such a notion a characterization in terms of induced c bundles is given for c dynamical systems such that the relative commutant of the fixed point c algebra is minimal i e it is generated by the centre of the given c algebra and the centre of the fixed point c algebra a class of examples in the setting of the cuntz algebra is given and connections with superselection structures with nontrivial centre are discussed

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

on cotriangular hopf algebras in an earlier paper we proved that any triangular semisimple hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k g of a finite group g by twisting its comultiplication by a twist in the sense of drinfeld in this paper we generalize this result to not necessarily finite dimensional cotriangular hopf algebras namely our main result says that a cotriangular hopf algebra a over k is obtained from a function algebra of a proalgebraic group by twisting its multiplication by a hopf 2 cocycle and possibly changing its r form by a central grouplike element of a of order 2 if and only if the trace of the squared antipode on any finite dimensional subcoalgebra of a is the dimension of this subcoalgebra the generalization like the original theorem is proved using deligne s theorem on tannakian categories we then give examples of twisted function algebras and in particular show that in the infinite dimensional case the squared antipode may not equal the identity on the other hand we show that in all of our examples the squared antipode is unipotent and conjecture it to be the case for any twisted function algebra we prove this conjecture in a large number of special cases using the quantization theory of the first author and d kazhdan

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

multi scenario modelling of uncertainty in stochastic chemical systems uncertainty analysis has not been well studied at the molecular scale despite extensive knowledge of uncertainty in macroscale systems the ability to predict the effect of uncertainty allows for robust control of small scale systems such as nanoreactors surface reactions and gene toggle switches however it is difficult to model uncertainty in such chemical systems as they are stochastic in nature and require a large computational cost to address this issue a new model of uncertainty propagation in stochastic chemical systems based on the chemical master equation is proposed in the present study the uncertain solution is approximated by a composite state comprised of the averaged effect of samples from the uncertain parameter distributions this model is then used to study the effect of uncertainty on an isomerization system and a two gene regulation network called a repressilator the results of this model show that uncertainty in stochastic systems is dependent on both the uncertain distribution and the system under investigation

on a certain approach to quantum homogeneous spaces we propose a definition of a quantum homogeneous space of a locally compact quantum group we show that classically it reduces to the notion of homogeneous spaces giving rise to an operator algebraic characterization of the transitive group actions on the quantum level our definition goes beyond the quotient case providing a framework which besides the vaes quotient of a locally compact quantum group by its closed quantum subgroup our main motivation is also compatible with generically non quotient quantum homogeneous spaces of a compact quantum group studied by p podle as well as the rieffel deformation of g homogeneous spaces finally our definition rules out the paradoxical examples of the non compact quantum homogeneous spaces of a compact quantum group

quantum isometries of the finite noncommutative geometry of the standard model we compute the quantum isometry group of the finite noncommutative geometry f describing the internal degrees of freedom in the standard model of particle physics we show that this provides genuine quantum symmetries of the spectral triple corresponding to m f where m is a compact spin manifold we also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries

nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales we present an efficient numerical algorithm for simulating chemical kinetic systems with multiple time scales this algorithm is an improvement of the traditional stochastic simulation algorithm ssa also known as gillespie s algorithm it is in the form of a nested ssa and uses an outer ssa to simulate the slow reactions with rates computed from realizations of inner ssas that simulate the fast reactions the algorithm itself is quite general and seamless and it amounts to a small modification of the original ssa our analysis of such multi scale chemical kinetic systems allows us to identify the slow variables in the system derive effective dynamics on the slow time scale and provide error estimates for the nested ssa efficiency of the nested ssa is discussed using these error estimates and illustrated through several numerical examples

presheaves of superselection structures in curved spacetimes we show that superselection structures on curved spacetimes that are expected to describe quantum charges affected by the underlying geometry are categories of sections of presheaves of symmetric tensor categories when an embedding functor is given the superselection structure is a tannaka type dual of a locally constant group bundle which hence becomes a natural candidate for the role of the gauge group indeed we show that any locally constant group bundle with suitable structure group acts on a net of c algebras fulfilling normal commutation relations on an arbitrary spacetime we also give examples of gerbes of c algebras defined by wightman fields and constructed using projective representations of the fundamental group of the spacetime which we propose as solutions for the problem that existence and uniqueness of the embedding functor are not guaranteed

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

hybrid framework for the simulation of stochastic chemical kinetics abstract stochasticity plays a fundamental role in various biochemical processes such as cell regulatory networks and enzyme cascades isothermal well mixed systems can be modelled as markov processes typically simulated using the gillespie stochastic simulation algorithm ssa 25 while easy to implement and exact the computational cost of using the gillespie ssa to simulate such systems can become prohibitive as the frequency of reaction events increases this has motivated numerous coarse grained schemes where the fast reactions are approximated either using langevin dynamics or deterministically while such approaches provide a good approximation when all reactants are abundant the approximation breaks down when one or more species exist only in small concentrations and the fluctuations arising from the discrete nature of the reactions become significant this is particularly problematic when using such methods to compute statistics of extinction times for chemical species as well as simulating non equilibrium systems such as cell cycle models in which a single species can cycle between abundance and scarcity in this paper a hybrid jump diffusion model for simulating well mixed stochastic kinetics is derived it acts as a bridge between the gillespie ssa and the chemical langevin equation for low reactant reactions the underlying behaviour is purely discrete while purely diffusive when the concentrations of all species are large with the two different behaviours coexisting in the intermediate region a bound on the weak error in the classical large volume scaling limit is obtained and three different numerical discretisations of the jump diffusion model are described the benefits of such a formalism are illustrated using computational examples

hybrid method for the chemical master equation the chemical master equation is solved by a hybrid method coupling a macroscopic deterministic description with a mesoscopic stochastic model the molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules the macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation the probability distribution is obtained by the stochastic simulation algorithm due to gillespie the equations are coupled via a summation over the mesoscale variables this summation is approximated by quasi monte carlo methods the error in the approximations is analyzed the hybrid method is applied to three chemical systems from molecular cell biology

the radial masa in free orthogonal quantum groups abstract we prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing and we compute the associated bimodule the proof relies on new properties of the jones wenzl projections and on an estimate of certain scalar products of coefficients of irreducible representations

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

solving chemical master equations by adaptive wavelet compression solving chemical master equations numerically on a large state space is known to be a difficult problem because the huge number of unknowns is far beyond the capacity of traditional methods we present an adaptive method which compresses the problem very efficiently by representing the solution in a sparse wavelet basis that is updated in each step the step size is chosen adaptively according to estimates of the temporal and spatial approximation errors numerical examples demonstrate the reliability of the error estimation and show that the method can solve large problems with bimodal solution profiles

galois theory for braided tensor categories and the modular closure abstract given a braided tensor category c with conjugate dual objects and irreducible unit together with a full symmetric subcategory s we define a crossed product c s this construction yields a tensor category with conjugates and an irreducible unit a category is a category enriched over vect c with positive operation a galois correspondence is established between intermediate categories sitting between c and c s and closed subgroups of the galois group gal c s c aut c c s of c the latter being isomorphic to the compact group associated with s by the duality theorem of doplicher and roberts denoting by d c the full subcategory of degenerate objects i e objects which have trivial monodromy with all objects of c the braiding of c extends to a braiding of c s iff s d under this condition c s has no non trivial degenerate objects iff s d if the original category c is rational i e has only finitely many isomorphism classes of irreducible objects then the same holds for the new one the category c c d is called the modular closure of c since in the rational case it is modular i e gives rise to a unitary representation of the modular group sl 2 z if all simple objects of s have dimension one the structure of the category c s can be clarified quite explicitly in terms of group cohomology

symmetries of quantum spaces subgroups and quotient spaces of quantum su 2 and so 3 groups we prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group we give the formula for the corresponding multiplicities in the case of the quotient quantum spaces we describe the subgroups and the quotient spaces of quantumsu 2 andso 3 groups

hybrid discrete continuum algorithms for stochastic reaction networks direct solutions of the chemical master equation cme governing stochastic reaction networks srns are generally prohibitively expensive due to excessive numbers of possible discrete states in such systems to enhance computational efficiency we develop a hybrid approach where the evolution of states with low molecule counts is treated with the discrete cme model while that of states with large molecule counts is modeled by the continuum fokker planck equation the fokker planck equation is discretized using a 2nd order finite volume approach with appropriate treatment of flux components the numerical construction at the interface between the discrete and continuum regions implements the transfer of probability reaction by reaction according to the stoichiometry of the system the performance of this novel hybrid approach is explored for a two species circadian model with computational efficiency gains of about one order of magnitude

higher dimensional algebra ii 2 hilbert spaces abstract a 2 hilbert space is a category with structures and properties analogous to those of a hilbert space more precisely we define a 2 hilbert space to be an abelian category enriched over hilb with a structure conjugate linear on the hom sets satisfying fg h g f h f hg we also define monoidal braided monoidal and symmetric monoidal versions of 2 hilbert spaces which we call 2 h algbebras braided 2 h algebras and symmetric 2 h algbebras and we describe the relation between these and tangles in two three and four dimensions respectively we prove a generalized doplicher roberts theorem starting that every symmetric 2 h algebra is equivalent to the category rep g of continuous unitary finite dimensional representations of some compact supergroupoid g the equivalence is given by a categorified version of the gelfand transform we also construct a categorified version of the fourier transform when g is a compact abelian group finally we characterize rep g by its universal properties when g is a compact classical group for example rep u n is the free connected symmetric 2 h algebra on one even object of dimension n

diagram projection rules for recoupling diagrams in the racah wigner category this paper gives three important techniques for recoupling coefficients of an arbitrary number or irreducible representations factorization as a product of smaller degree recoupling coefficients diagram projection projecting a recoupling diagram to obtain constraints on recoupling coefficients and primitive expansion expressing a recoupling coefficient in terms of primitive recoupling coefficients the first technique reduces a recoupling coefficient to its essential and simplest form the second is many faceted allowing one to work at the recoupling level importantly recoupling diagrams can be used to construct a complete set of constraints on the recoupling coefficients the third allows large degree recoupling coefficients to be expressed in terms of primitive recoupling coefficients an example of this is the fact that the 9j symbol is defined by the 3j and 6j symbols

quantum rigidity of negatively curved manifolds we show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected riemannian manifold m g with strictly negative curvature is automatically classical in the sense that it factors through the action of the isometry group of m g this partially answers a question by d goswami

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

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

liberation of orthogonal lie groups abstract we show that under suitable assumptions we have a one to one correspondence between classical groups and free quantum groups in the compact orthogonal case we classify the groups under correspondence with the result that there are exactly 6 of them o n s n h n b n s n b n we investigate the representation theory aspects of the correspondence with the result that for o n s n h n b n this is compatible with the bercovici pata bijection finally we discuss some more general classification problems in the compact orthogonal case notably with the construction of a new quantum group

compact quantum metric spaces and ergodic actions of compact quantum groups we show that for any co amenable compact quantum group a c g there exists a unique compact hausdorff topology on the set ea g of isomorphism classes of ergodic actions of g such that the following holds for any continuous field of ergodic actions of g over a locally compact hausdorff space t the map t ea g sending each t in t to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of in each fibre is continuous over t for every equivalence class of irreducible unitary representations of g generalizations for arbitrary compact quantum groups are also obtained in the case g is a compact group the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of landstad and wassermann podle spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum su 2 group we also introduce a notion of regularity for quantum metrics on g and show how to construct a quantum metric from any ergodic action of g starting from a regular quantum metric on g furthermore we introduce a quantum gromov hausdorff distance between ergodic actions of g when g is separable and show that it induces the above topology

an invariant for homogeneous spaces of compact quantum groups abstract the central notion in connes formulation of noncommutative geometry is that of a spectral triple given a homogeneous space of a compact quantum group restricting our attention to all spectral triples that are well behaved with respect to the group action we construct a certain dimensional invariant in particular taking the quantum group itself as the homogeneous space this gives an invariant for a compact quantum group computations of this invariant in several cases including all type a quantum groups are given

the baum connes conjecture for free orthogonal quantum groups we prove an analogue of the baum connes conjecture for free orthogonal quantum groups more precisely we show that these quantum groups have a element and that 1 it follows that free orthogonal quantum groups are k amenable we compute explicitly their k theory and deduce in the unimodular case that the corresponding reduced c algebras do not contain nontrivial idempotents our approach is based on the reformulation of the baum connes conjecture by meyer and nest using the language of triangulated categories an important ingredient is the theory of monoidal equivalence of compact quantum groups developed by bichon de rijdt and vaes this allows us to study the problem in terms of the quantum group suq 2 the crucial part of the argument is a detailed analysis of the equivariant kasparov theory of the standard podle

superselection theory for subsystems an inclusion of observable nets satisfying duality induces an inclusion of canonical field nets any bose net intermediate between the observable net and the field net and satisfying duality is the fixed point net of the field net under a compact group this compact group is its canonical gauge group if the occurrence of sectors with infinite statistics can be ruled out for the observable net and its vacuum hilbert space is separable

canonical actions on o we show that the group of automorphisms of the cuntz algebra o canonically associated to a group g of unitaries on the generating hilbert space is ergodic if and only if g has no non zero finite dimensional stable subspace it is prime if g is strongly compact and more generally we give necessary and sufficient conditions if the defining representation of g is the direct sum of finite dimensional representations with determinant one the fixed point subalgebra og of o is a simple c algebra these results extend in part to o results previously known for on n

a duality theorem for ergodic actions of compact quantum groups on c algebras the spectral functor of an ergodic action of a compact quantum group g on a unital c algebra is quasitensor in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation and the inclusion maps satisfy natural properties we show that any quasitensor functor from rep g to the category of hilbert spaces is the spectral functor of an ergodic action of g on a unital c algebra

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

topological aperiodicity for product systems over semigroups of ore type we prove a version of uniqueness theorem for cuntz pimsner algebras of discrete product systems over semigroups of ore type to this end we introduce doplicher roberts picture of cuntz pimsner algebras and the semigroup dual to a product system of regular c correspondences under a certain aperiodicity condition on the latter we obtain the uniqueness theorem and a simplicity criterion for the algebras in question these results generalize the corresponding ones for crossed products by discrete groups due to archbold and spielberg and for exel s crossed products due to exel and vershik they also give interesting conditions for topological higher rank graphs and p graphs and apply to the new cuntz c algebra qn arising from the ax b semigroup over 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

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

completely positive maps of the cuntz algebras we construct a covariant functor from the category whose objects are the complex infinite dimensional separable hilbert spaces and whose morphisms are the contractions into the category whose objects are the unitalc algebras and whose morphisms are the completely positive identity preserving maps thec algebras involved are isomorphic to the cuntz algebra o and our functor extends the canonical action of the group of unitaries on the generating hilbert space our result cannot apply to on n

ergodic actions of universal quantum groups on operator algebras we construct ergodic actions of compact quantum groups on c algebras and von neumann algebras and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups

bundles of c categories ii c dynamical systems and dixmier douady invariants we introduce a cohomological invariant arising from a class in nonabelian cohomology this invariant generalises the dixmier douady class and encodes the obstruction to a c algebra bundle being the fixed point algebra of a gauge action as an application the duality breaking for group bundles vs tensor c categories with nonsimple unit is discussed in the setting of nistor troitsky gauge equivariant k theory there is a map assigning a nonabelian gerbe to a tensor category and triviality of the gerbe is equivalent to the existence of a dual group bundle at the c algebraic level this corresponds to studying c algebra bundles with fibre a fixed point algebra of the cuntz algebra and in this case our invariant describes the obstruction to finding an embedding into the cuntz pimsner algebra of a vector bundle

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

an adaptive wavelet method for the chemical master equation an adaptive wavelet method for the chemical master equation is constructed the method is based on the representation of the solution in a sparse haar wavelet basis the time integration by rothe s method and an iterative procedure which in each time step selects those degrees of freedom which are essential for propagating the solution the accuracy and efficiency of the approach is discussed and the performance of the adaptive wavelet method is demonstrated by numerical examples

conjugation in representation categories of multiplicative unitaries and their actions on c algebras abstract a conjugation functor f on a full subcategory of r v the representation category of a multiplicative unitaryv is defined ifvhas a conjugate it is also regular and the domain of f is all r v examples of selfconjugate multiplicative unitaries are discussed a coaction of the hopfc algebra associated withvon the cuntz algebra o dis canonically defined by a unitary objectwof r v acting on ad dimensional hilbert space as in the group action case ifd andwbelongs to the domain of f ergodic coactions are often characterized by the absence of finite dimensional subrepresentations ofw furthermore model actions of compact quantum group duals onc algebras are defined

highly entangled non random subspaces of tensor products from quantum groups in this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional hilbert spaces arising from the representation theory of free orthogonal quantum groups we determine their largest singular values and obtain lower bounds for the minimum output entropy of the corresponding quantum channels an application to the construction of d positive maps on matrix algebras is also presented

quantum charges and spacetime topology the emergence of new superselection sectors a new form of superselection sectors of topological origin is developed by that it is meant a new investigation that includes several extensions of the traditional framework of doplicher haag and roberts in local quantum theories at first we generalize the notion of representations of nets of c algebras then we provide a brand new view on selection criteria by adopting one with a strong topological flavour we prove that it is coherent with the older point of view hence a clue to a genuine extension in this light we extend roberts cohomological analysis to the case where 1 cocycles bear non trivial unitary representations of the fundamental group of the spacetime equivalently of its cauchy surface in the case of global hyperbolicity a crucial tool is a notion of group von neumann algebras generated by the 1 cocycles evaluated on loops over fixed regions one proves that these group von neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type i all that amounts to a new invariant in a topological sense which can be defined as the dimension of the factor we prove that any 1 cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored this second part much resembles what in literature is known as geometric phases indeed by the very geometrical origin of the 1 cocycles that we discuss in the paper they are essential tools in the theory of net bundles and the topological part is related to their holonomy content at the end we prove the existence of net representations

quantum isometry groups of the podles spheres abstract for 0 1 c 0 we identify the quantum group so 3 as the universal object in the category of compact quantum groups acting by orientation and volume preserving isometries in the sense of bhowmick and goswami 2009 4 on the natural spectral triple on the podles sphere s c 2 constructed by dabrowski d andrea landi and wagner 2007 in 9