operators preserving orthogonality are isometries we prove that every operator preserving orthogonality in a real banach space is an isometry multiplied by a constant

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

mappings approximately preserving orthogonality in normed spaces abstract we answer many open questions regarding approximately orthogonality preserving mappings in birkhoff james sense in normed spaces in particular we show that every approximately orthogonality preserving linear mapping in chmielinski sense is necessarily a scalar multiple of an e isometry thus whenever e isometries are close to isometries we obtain stability an example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries that is stability does not hold

mappin class group actions on quantum doubles we study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non semisimple hopf algebra that arise in the construction of universal extended topological field theories we discuss how for doubles the degeneracy problem of tqft s is circumvented we find compact formulae for thes 1 matrices using the canonical non degenerate forms of hopf algebras and the bicrossed structure of doubles rather than monodromy matrices a rigorous proof of the modular relations and the computation of the projective phases is supplied using radford s relations between the canonical forms and the moduli of integrals we analyze the projectivesl 2 z action on the center ofuq sl2 forq anl 2m 1st root of unity it appears that the 3m 1 dimensional representation decomposes into anm 1 dimensional finite representation and a2m dimensional irreducible representation the latter is the tensor product of the two dimensional standard representation ofsl 2 z and the finite m dimensional representation obtained from the truncated tqft of the semisimplified representation category ofuq sl2

categories premodulaires modularisations et invariants des varietes de dimension 3 an abelian k linear semisimple category having a finite number of simple objects and endowed with a ribbon structure is called premodular it is modular in the sense of turaev if the so called s matrix is invertible a modular category defines invariants of 3 manifolds and a tqft t when is it possible to construct a modularisation of a given premodular category i e a functor to a modular category preserving the structures and dominant in a certain sense it turns out 2 3 that this amounts essentially to making transparent objects trivial we give a full answer to this problem in the case when k is a field of char 0 as well as partial answers in char p under a few obvious hypotheses a premodular category admits a modularisation which is unique th 3 1 and cor 3 5 in char 0 the proof relies on two main ingredients a new and very simple criterion for the s matrix to be invertible 1 1 and deligne s internal characterization of tannakian categories in char 0 d when simple transparent objects are invertible the criterion is simpler 4 2 and the modularisation can be described more explicitly prop 4 4 we conclude with two examples the premodular categories associated with quantum hbox sf sl n and hbox sf pgl n at roots of unity in the first case we obtain modular categories which were built independently by c blanchet b in the second case we obtain modularizations in all the cases where y yokota y found reshetikhin turaev invariants of 3 manifolds thereby improving as well as explaining yokota s results

virasoro frames and their stabilizers for the e8 lattice type vertex operator algebra the concept of a framed vertex operator algebra fvoa is new cf dgh this article contributes to this theory with a full analysis of all virasoro frame stabilizers in v the important example of the e8 level 1 affine kac moody voa which is isomorphic to the lattice voa for the root lattice of e8 c we analyze the frame stabilizers both as abstract groups and as subgroups of aut v e8 c each frame stabilizer is a finite group contained in the normalizer of a 2b pure elementary abelian 2 group in aut v but is not usually a maximal finite subgroup of this normalizer in particular we prove that there are exactly five orbits for the action of aut v on the set of virasoro frames thus settling an open question about v in section 5 of dgh the results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories so this article contributes also to this theory there are two main viewpoints in our analysis the first is the theory of codes lattices markings and the resulting groups of automorphisms the second is the theory of finite subgroups of lie groups we expect our methods to be applicable to the study of other fvoas and their frame stabilizers appendices present aspects of the theory of automorphism groups of voas in particular there is a general result of independent interest on embedding lattices into unimodular lattices so as to respect automorphism groups and definiteness

crossed s matrices and fourier matrices for coxeter groups with automorphism abstract we study crossed s matrices for braided g crossed categories and reduce their computation to a submatrix of the de equivariantization we study the more general case of a category containing the symmetric category rep a z with a a finite cyclic group and z a such that z 2 1 we give two examples of such categories which enable us to recover the fourier matrix associated with the big family of unipotent characters of the dihedral groups with automorphism as well as the fourier matrix of the big family of unipotent characters of the ree group of type f 4 2

invariants of spin three manifolds from chern simons theory and finite dimensional hopf algebras abstract a version of kirby calculus for spin and framed three manifolds is given and is used to construct invariants of spin and framed three manifolds in two situations the first is ribbon categories which possess odd degenerate objects this case includes the quantum group situations corresponding to the half integer level chern simons theories conjectured to give spin tqfts by dijkgraaf and witten 1990 commun math phys 129 393 429 in particular the spin invariants constructed by kirby and melvin 1991 invent math 105 473 545 are shown to be identical to the invariants associated to so 3 second an invariant of spin manifolds analogous to the hennings invariant is constructed beginning with an arbitrary factorizable unimodular quasitriangular hopf algebra in particular a framed manifold invariant is associated to every finite dimensional hopf algebra via its quantum double and is conjectured to be identical to kuperberg s noninvolutory invariant of framed manifolds associated to that hopf algebra

sl 2 z action for ribbon quasi hopf algebras abstract we study the universal hopf algebra l of majid and lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi hopf algebra a we show that l a with coadjoint action and compute the hopf algebra structure morphisms of l in terms of the defining data of a we give explicitly the condition on a which makes rep a factorisable and compute lyubashenko s projective s l 2 z action on the centre of a in this case the point of this exercise is to provide the groundwork for the applications to ribbon categories arising in logarithmic conformal field theories in particular symplectic fermions and w p models and to test a conjectural non semisimple verlinde formula

linear mappings approximately preserving orthogonality abstract we consider the class of linear mappings between real or complex inner product spaces such that for each two orthogonal vectors in the domain their values are orthogonal in the target space by orthogonality 0 1 we mean the relation u v u v u v

invariant rings through categories abstract we formulate a notion of geometric reductivity in an abstract categorical setting which we refer to as adequacy the main theorem states that the adequacy condition implies that the ring of invariants is finitely generated this result applies to the category of modules over a bialgebra the category of comodules over a bialgebra and the category of quasi coherent sheaves on an algebraic stack of finite type over an affine base

higher genus mapping class group invariants from factorizable hopf algebras abstract lyubashenko s construction associates representations of mapping class groups map g n of riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category we consider this construction as applied to the category of bimodules over a finite dimensional factorizable ribbon hopf algebra h for any such hopf algebra we find an invariant of map g n for all values of g and n more generally we obtain such invariants for any pair h where is a ribbon automorphism of h our results are motivated by the quest to understand higher genus correlation functions of bulk fields in two dimensional conformal field theories with chiral algebras that are not necessarily semisimple so called logarithmic conformal field theories

an identity for δ e approximately orthogonality preserving mappings abstract suppose t is a hilbert space operator given 0 1 we define e t to be the smallest e for which t is e approximately orthogonality preserving and then obtain an exact formula for e t in terms of t and the minimum modulus m t of t for two nonzero operators t s it follows from the formula that t is e s e aop if and only if s is e t e aop where e t e 0 t finally we show that an operator t is e aop if and only if there exists a special aop operator s such that ts is e aop theorem 3 8

schur weyl quasi duality and co triangular hopf quasigroups schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic lie theory in this paper we will introduce the notion of schur weyl quasi duality and study it for this generally we consider a braided vector space v c and its braided lie algebra endk v then we can construct its braided enveloping algebra u endk v which is a connected braided c cocommutative hopf algebra let h be a triangular hopf quasigroup with bijective antipode and b be a cotriangular hopf quasigroup with bijective antipode let v be any finite dimensional vector space in the category lq h r b of generalized long quasimodules we show that u endkv h b ksn v n is a schur weyl quasi duality under suitable conditions

on the classification of the grothendieck rings of non self dual modular categories abstract we develop a symbolic computational approach to classifying low rank modular fusion categories up to finite ambiguity by a generalized form of ocneanu rigidity due to etingof ostrik and nikshych it is enough to classify modular fusion algebras of a given rank that is to determine the possible grothendieck rings with modular realizations we use this technique to classify modular categories of rank at most 5 that are non self dual i e those for which some object is not isomorphic to its dual object

finite size dependence of the helicity modulus within the mean spherical model the validity of the finite size scaling prediction about the existence of logarithmic corrections in the helicity modulus of three dimensional o n symmetric order parameter systems in confined geometries is studied for the three dimensional mean spherical model of geometryl 3 s d d 0 d 3 for a fully finite geometry the general case ofd p 0 periodic d a 0 antiperiodic d 0 0 free andd 1 0 fixed d p da d0 d1 d d 3 boundary conditions is considered whereas for film d 2 and cylinder d 1 geometries only the case of antiperiodic and or periodic boundary conditions is investigated the corresponding expressions for the finite size scaling function of the helicity modulus and its asymptotics in the vicinity below and above the bulk critical temperaturet c and the shifted critical temperaturet c l are derived the obtained results are not in agreement with the hypothesis of the existence of a log l correction term to the finite size behavior of the helicity modulus in the finite size critical region if d 3 in the case of film and cylinder geometries there are no logarithmic corrections in the case of a fully finite geometry a universal logarithmic correction term d 0 d 1 4 2da 1 2 lnl l is obtained only for t c t l lnl

a spin decomposition of the verlinde formulas for type a modular categories a modular category is a braided category with some additional algebraic features the interest of this concept is that it provides a topological quantum field theory in dimension 3 the verlinde formulas associated with a modular category are the dimensions of the tqft modules we discuss reductions and refinements of these formulas for modular categories related with su n our main result is a splitting of the verlinde formula corresponding to a brick decomposition of the tqft modules whose summands are indexed by spin structures modulo an even integer we introduce here the notion of a spin modular category and give the proof of the decomposition theorem in this general context

quasitriangular g cograded multiplier hopf algebras we put the known results on the antipode of a usual quasitriangular hopf algebra into the framework of multiplier hopf algebras we illustrate with examples which cannot be obtained by using classical hopf algebras the focus of the present paper lies on the class of the so called g cograded multiplier hopf algebras by doing so we bring the results of quasitriangular hopf group coalgebras as introduced by turaev to the more general framework of multiplier hopf algebras

on phase separation in the generalised spherical model it is proved that the interface in the generalised spherical model the d to infinity limit of the d vector model with kac helfand type interactions is diffuse at all temperatures the interface width is shown to be proportional to the thickness of the sample the magnetisation profile is also obtained

coribbon hopf face algebras generated by lattice models abstract by studying points of the underlying quantum groups of coquasitriangular hopf face algebras we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type also we give a classification of the braiding and the ribbon structure on quantized classical groups and modular tensor categories closely related to quantum su n l invariants of 3 manifolds

renormalized hennings invariants and 2 1 tqfts we construct non semisimple 2 1 tqfts yielding mapping class group representations in lyubashenko s spaces in order to do this we first generalize beliakova blanchet and geer s logarithmic hennings invariants based on quantum mathfrak sl 2 to the setting of finite dimensional non degenerate unimodular ribbon hopf algebras the tools used for this construction are a hennings augmented reshetikhin turaev functor and modified traces when the hopf algebra is factorizable we further show that the universal construction of blanchet habegger masbaum and vogel produces a 2 1 tqft on a not completely rigid monoidal subcategory of cobordisms

invariants of 3 manifolds and projective representations of mapping class groups via quantum groups at roots of unity an example of a finite dimensional factorizable ribbon hopf algebra is given by a quotienth u q g of the quantized universal enveloping algebrau q g at a root of unityq of odd degree the mapping class groupm g 1 of a surface of genusg with one hole projectively acts by automorphisms in theh moduleh g ifh is endowed with the coadjointh module structure there exists a projective representation of the mapping class groupm g n of a surface of genusg withn holes labeled by finite dimensionalh modulesx 1 x n in the vector space hom h x 1 x n h g an invariant of closed oriented 3 manifolds is constructed modifications of these constructions for a class of ribbon hopf algebras satisfying weaker conditions than factorizability including most ofu q g at roots of unityq of even degree are described

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

approximately orthogonality preserving mappings on c modules abstract we study orthogonality preserving and approximately orthogonality preserving mappings in the setting of inner product c modules in particular if v and w are inner product c modules over the c algebra a any scalar multiple of an a linear isometry is an a linear orthogonality preserving mapping the converse does not hold in general but it holds if a contains k h the c algebra of all compact operators on a hilbert space h furthermore we give the estimate of t x t y t 2 x y for an a linear approximately orthogonality preserving mapping t v w when v and w are inner product c modules over a c algebra containing k h in the case a k h and v w are hilbert we also prove that an a linear approximately orthogonality preserving mapping can be approximated by an a linear orthogonality preserving mapping

on mappings approximately preserving orthogonality abstract we answer a question posed by chmielinski whether a linear map which approximately preserves orthogonality must be close to an orthogonality preserving one furthermore we give a short proof of the stability of the orthogonality equation on finite dimensional hilbert spaces

on the central charge of a factorizable hopf algebra abstract for a semisimple factorizable hopf algebra over a field of characteristic zero we show that the value that an integral takes on the inverse drinfel d element differs from the value that it takes on the drinfel d element itself by at most a fourth root of unity this can be reformulated by saying that the central charge of the hopf algebra is an integer if the dimension of the hopf algebra is odd we show that these two values differ by at most a sign which can be reformulated by saying that the central charge is even we give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs to formulate our results we use the language of modular data

isosceles orthogonality preserving property and its stability for real normed spaces we consider the class of linear operators preserving approximately the relation of isosceles orthogonality we show some general properties of such mappings next we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one in regard to this we generalize some results obtained earlier for inner product spaces with standard orthogonality relation

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

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

from non semisimple hopf algebras to correlation functions for logarithmic cft we use factorizable finite tensor categories and specifically the representation categories of factorizable ribbon hopf algebras h as a laboratory for exploring bulk correlation functions in local logarithmic conformal field theories for any ribbon hopf algebra automorphism of h we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric frobenius algebra we derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters as a crucial ingredient this involves the cartan matrix of the category we also show how for any candidate bulk state space of the type we consider correlation functions of bulk fields for closed oriented world sheets of any genus can be constructed that are invariant under the natural action of the relevant mapping class group

a geometrical presentation of the surface mapping class group and surgery we construct a tangle presentation of the mapping class group similar to a natural presentation of the braid group by geometrical braids a relation between surgery and heegard diagrams for 3 manifolds arising in this way and different applications are studied

further results on the critical behaviour of the mean spherical model layer and local susceptibilities abstract the critical behaviour of some properties of the ferromagnetic mean spherical model with a film geometry l 2 and the neumann dirichlet boundary conditions is investigated in the presence of three external fields a bulk field a step like field which changes sign at distance lx 0 x 1 from the neumann boundary and a surface field acting on the lth layer the behaviour of the mean spherical constraint is studied at three different temperatures and field regimes high temperature bulk limit critical finite size scaling regime and low temperature moderate field regime exact expressions for the layer and local susceptibilities are derived and studied in the three different regimes

new effects in the magnetization profile of the spherical model in a step like field the ferromagnetic mean spherical model with a layer geometry of thickness l and neumann dirichlet boundary conditions is investigated in the presence of a step like external field which changes sign at distance from the neumann boundary the amplitude of the field can be taken to vanish in the thermodynamic limit as an inverse power of l exact expressions for the magnetization profile are derived and studied in three different temperature and field regimes high temperature bulk limit critical finite size scaling regime and low temperature moderate field regime it is found that in the critical finite size scaling regime there exist two special values of x denoted by which depend on the scaled temperature and field variables and have the property that the magnetization changes sign only when the magnetization is everywhere negative when and everywhere positive when in the low temperature moderate field regime we establish that the field induced critical point in the case of periodic boundary conditions and a step like field with appears at

new surface critical exponents in the spherical model the three dimensional mean spherical model with a l layer film geometry under neumann neumann and neumann dirichlet boundary conditions is considered surafce fields and are supposed to act at the surfaces bounding the system in the case of neumann boundary conditions a new surface critical exponent is found it is argued that this exponent corresponds to a special surface bulk phase transition in the model the privman fisher scaling hypothesis for the free energy is verified and the corresponding scaling functions for both the neumann neumann and neumann dirichlet boundary conditions are explicitly derived if the layer field is applied at some distance from the dirichlet boundary a family of critical exponents emerges their values depend on the exponent defining how the distance scales with the finite size of the system and interpolate continuously between the extreme cases and

kirby elements and quantum invariants we define the notion of a kirby element of a ribbon category mathcal c not necessarily semisimple kirby elements lead to 3 manifold invariants we characterize a class of kirby elements the algebraic kirby elements in terms of the structure maps of a hopf algebra in mathcal c this class is sufficiently large to recover the quantum invariants of 3 manifolds of reshetikhin and turaev of hennings kauffman and radford and of lyubashenko when these are well defined the cases of a semisimple ribbon category and of a category of representations are explored in detail

surface critical exponents for a three dimensional modified spherical model a modified three dimensional mean spherical model with a l layer film geometry under neumann neumann boundary conditions is considered two spherical fields are present in the model a surface one fixes the mean square value of the spins at the boundaries at some 0 and a bulk one imposes the standard spherical constraint the mean square value of the spins in the bulk equals one the surface susceptibility 1 1 has been evaluated exactly for 1 we find that 1 1 is finite at the bulk critical temperature tc in contrast with the recently derived value 1 1 1 in the case of just one global spherical constraint the result 1 1 1 is recovered only if c 2 12kc 1 where kc is the dimensionless critical coupling when c 1 1 diverges exponentially as t t c an effective hamiltonian which leads to an exactly solvable model with 1 1 2 the value for the n limit of the corresponding o n model is proposed too

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

aop mappings and the distance to the scalar multiples of isometries in this paper linear e orthogonality preserving mappings are studied we define e t as the smallest e for which t is e orthogonality preserving and then derive an exact formula for e t in terms of t and the minimum modulus m t of t we see that e orthogonality preserving mappings for some e 1 are exactly the operators that are bounded from below we improve an upper bound in the stability equation given in 7 theorem 2 3 which was thought to be sharp

stability of the orthogonality preserving property in finite dimensional inner product spaces we establish a stability property for inner product preserving not necessarily linear mappings then as a consequence we show that a linear mapping defined on a finite dimensional inner product space which approximately preserves orthogonality can be approximated by a linear orthogonality preserving one

non degeneracy conditions for braided finite tensor categories abstract for a braided finite tensor category c with unit object 1 c lyubashenko considered a certain hopf algebra f c endowed with a hopf pairing f f 1 to define the notion of a non semisimple modular tensor category we say that c is non degenerate if the hopf pairing is non degenerate in this paper we show that c is non degenerate if and only if it is factorizable in the sense of etingof nikshych and ostrik if and only if its muger center is trivial if and only if the linear map hom c 1 f hom c f 1 induced by the pairing is invertible as an application we prove that the category of yetter drinfeld modules over a hopf algebra in c is non degenerate if and only if c is

the group of automorphisms of a semisimple hopf algebra over a field of characteristic 0 is finite introduction suppose that a is a semisimple hopf algebra over a field of characteristic 0 or that a is a semisimple cosemisimple involutory hopf algebra over a field k of characteristic p dim a in this paper we prove that the group authopf a of hopf algebra automorphisms of a is finite we show that the semigroup endhopf a of hopf algebra endomorphisms of a is also finite as a corollary generally the group of hopf algebra automorphisms of a finite dimensional hopf algebra need not be finite for any positive integer n and any field k we construct a family of finite dimensional hopf algebras over k with automorphism group gln k the fact that authopf a is finite is a consequence of a theorem of

modular invariant frobenius algebras from ribbon hopf algebra automorphisms abstract for any finite dimensional factorizable ribbon hopf algebra h and any ribbon automorphism of h we establish the existence of the following structure an h bimodule f and a bimodule morphism z from lyubashenko s hopf algebra object k for the bimodule category to f this morphism is invariant under the natural action of the mapping class group of the one punctured torus on the space of bimodule morphisms from k to f we further show that the bimodule f can be endowed with a natural structure of a commutative symmetric frobenius algebra in the monoidal category of h bimodules and that it is a special frobenius algebra iff h is semisimple the bimodules k and f can both be characterized as coends of suitable bifunctors the morphism z is obtained by applying a monodromy operation to the coproduct of f a similar construction for the product of f exists as well our results are motivated by the quest to understand the bulk state space and the bulk partition function in two dimensional conformal field theories with chiral algebras that are not necessarily semisimple

radford s s 4 formula for co frobenius hopf algebras abstract this note extends radford s formula for the fourth power of the antipode of a finite dimensional hopf algebra to co frobenius hopf algebras and studies equivalent conditions to a hopf algebra being involutory for finite dimensional and co frobenius hopf algebras

linear mappings preserving ρ orthogonality abstract we define a orthogonality in a real normed space and we consider the class of linear mappings preserving this relation we show that a linear mapping preserving orthogonality has to be a similarity i e a scalar multiple of an isometry as a result we give a characterization of smooth spaces in terms of this orthogonality

jones witten invariants for nonsimply connected lie groups and the geometry of the weyl alcove abstract the quotient process of muger and bruguieres is used to construct modular categories and tqfts out of closed subsets of the weyl alcove of a simple lie algebra in particular it is determined at which levels closed subsets associated to nonsimply connected groups lead to tqfts many of these tqfts are shown to decompose into a tensor product of tqfts coming from smaller subsets the prime subsets among these are classified and apart from some giving tqfts depending on homology as described by murakami ohtsuki and okada they are shown to be in one to one correspondence with the tqfts predicted by dijkgraaf and witten to be associated to chern simons theory with a nonsimply connected lie group thus in particular a rigorous construction of the dijkgraaf witten tqfts is given as a byproduct a purely quantum groups proof of the modularity of the full weyl alcove for arbitrary quantum groups at arbitrary levels is given

on mappings approximately transferring relations in finite dimensional normed spaces abstract we discuss the problem of stability of relations preserving property for finite dimensional normed spaces moreover we obtain similar results for other orthogonality relations next we show a counterexample for infinite dimensional normed spaces

correspondences of ribbon categories much of algebra and representation theory can be formulated in the general framework of tensor categories the aim of this paper is to further develop this theory for braided tensor categories several results are established that do not have a substantial counterpart for symmetric tensor categories in particular we exhibit various equivalences involving categories of modules over algebras in ribbon categories finally we establish a correspondence of ribbon categories that can be applied to and is in fact motivated by the coset construction in conformal quantum field theory

duals of quasitriangular hopf superalgebras and the classical limit the duality of quasitriangular hopf superalgebras and superbialgebras with so called main commutation relations mcr which is established by means of bilinear forms with values in a finite dimensional representation of the universal r matrix r is used to calculate relations of quantum supercoordinates for several examples the tensor product of representations of r with respect to the comultiplication then leads to the construction of an accordingly deformed tensor product of mcr type superbialgebras the correspondingly deformed tensor product of quantum vector superspaces is then established in the classical limit of lie superbialgebras one similarly obtains supercommutation relations which may be understood in case of quantum groups with deformation parameter q 1 as mcr in the first order of ln q again the deformed tensor product of such semiclassical mcr type lie superbialgebras is constructed

the small quantum group as a quantum double we prove that the quantum double of the quasi hopf algebra aq g of dimension ndimg attached in p etingof s gelaki on radically graded finite dimensional quasi hopf algebras mosc math j 5 2 2005 371 378 to a simple complex lie algebra g and a primitive root of unity q of order n2 is equivalent to lusztig s small quantum group uq g under some conditions on n we also give a conceptual construction of aq g using the notion of de equivariantization of tensor categories

on the antipode of a quasitriangular hopf algebra let a r be a quasitriangular hopf algebra with antipode s in the category of vector spaces over a field k in this paper we show that s2 is inner which means s is bijective and when a is finite dimensional there is a grouplike element h e a such that s4 a h r for all a e a suppose that a is any hopf algebra with antipode s over the field k proposition 1 of section 1 gives a sufficient condition for 3 to be inner which we apply to quasitriangular hopf algebras in section 2 its statement and proof are based on calculations found in 2 pp 66671 generally s2 need not be inner suppose that a is finite dimensional if s2 were inner then all ideals of a would be invariant under s2 and hence all subcoalgebras of the dual hopf algebra a would be invariant under s where s s is the antipode of a there are examples 5 over algebraically closed lields in which the latter is not the case section 2 begins with a discussion of properties of quasit angular hopf algebras a r over the held k used in this paper write r cr c r ea a and set u cs r r the main result of section 2 is that u is invertible and s2 a uau for all a e a this result was established in z exercise 7 3 61 under the hypothesis that s is bijective if a is any finite dimensional hopf algebra with antipode s over the field k there are distinguished grouplike elements g e a and tl e a which relate left and right integrals and which together describe s4 in section 3 we consider s4 when a r is a unite dimensional quasit angular hopf algebra let h iiu where u c s r r l is defined as above and u s u i by relating h to g and 1 we are able to show that h is a grouplike element and that s4 a hah for all a e a when a is unimodular we prove that h g thus the product vu does not depend on r in the unimodular case we assume that the reader has some familiarity with the elementary aspects of the theory of hopf algebras a recommended reference is 6

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

the influence of external boundary conditions on the spherical model of a ferromagnet i magnetization profiles the spherical model of a ferromagnet is investigated for various external boundary conditions it is shown that besides the well known critical point a second one can be produced by the boundary conditions although the main asymptotic of the free energy is analytic at the new critical point theo n1 2 d asymptotic possesses a singularity here a natural order parameter of the model has singularities at both critical points the magnetization profile is studied for the whole range of the model s parameters and at different scales it is shown that in an appropriate regime below the second critical temperature the magnetization profile freezes that is becomes temperature independent distributions of the single spin variables and some macroscopic observables including normalized total spin are studied for the whole temperature range including the critical points

generalized stability of isometries abstract let x and y be real banach spaces and let e p 0 a mapping f x y is called an e p isometry if f x f y x y e x y p holds for all x y x a pair x y is p stable with respect to isometries if there exists a function 0 0 with lime 0 e 0 such that for every surjective e p isometry f x y there is a surjective isometry u x y satisfying the estimate f x u x e x p x x we show that every pair of banach spaces x y is p stable for 0 p r 2 r 2 is not 1 stable when p 1 a superstability phenomenon occurs for finite dimensional banach spaces

dichromatic state sum models for four manifolds from pivotal functors a family of invariants of smooth oriented four dimensional manifolds is defined via handle decompositions and the kirby calculus of framed link diagrams the invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category a state sum formula for the invariant is constructed via the chain mail procedure so a large class of topological state sum models can be expressed as link invariants most prominently the crane yetter state sum over an arbitrary ribbon fusion category is recovered including the nonmodular case it is shown that the crane yetter invariant for nonmodular categories is stronger than signature and euler invariant a special case is the four dimensional untwisted dijkgraaf witten model derivations of state space dimensions of tqfts arising from the state sum model agree with recent calculations of ground state degeneracies in walker wang models relations to different approaches to quantum gravity such as cartan geometry and teleparallel gravity are also discussed

braided groups and quantum fourier transform abstract we show that acting on every finite dimensional factorizable ribbon hopf algebra h there are invertible operators s t obeying the modular identities s t 3 s 2 where is a constant the class includes the finite dimensional quantum groups uq g associated to complex simple lie algebras we give the example of uq sl 2 at a root of unity in detail as well as an example relating to anyons the operator s plays the role of quantum fourier transform and acts naturally on h viewed by transmutation as a braided group h a braided cocommutative hopf algebra in a braided category it obeys s 2 s 1 where s is the antipode of h the results follow as an application of previous category theoretical constructions

on certain basis connected with operator and its applications abstract we assume that h k are the complex hilbert spaces and a linear operator t b h k is given the object of this paper is to define and consider the certain basis of h connected with the operator t b h k this basis will play an important role in describing the approximately orthogonality preserving operators next we will give some application of our results we give new results and more general ones namely we answer a question whether a linear operator which approximately preserves orthogonality must be close to an orthogonality preserving one