conformal invariance and finite one dimensional quantum chains based on previous work of cardy 1984 the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions the method is illustrated in the case of the three and four state potts models where the anomalous dimensions of the para fermionic operators are found

a renormalisation group analysis of the semi finite potts model we discuss the ordinary transition of the semi infinite q state potts model within the migdal kadanoff renormalisation group scheme approximate surface free energies are calculated in dimension d 2 in addition we consider the effect of symmetry breaking surface fields our scheme predicts that only one such field is relevant

weak c cleft extensions weak entwining structures and weak hopf algebras abstract we formulate the concept of weak cleft extension for a weak entwining structure in a braided monoidal category c with equalizers and coequalizers we prove that if a is a weak c cleft extension then there is an isomorphism of algebras between a and a subobject of the tensor product of a c and c where a c is a subalgebra of a also we prove the corresponding dual results and linking the information of this two parts we obtain a general property for a pair morphisms f c a and g a c of algebras and coalgebras satisfying certain conditions finally as particular instances we get the results of fernandez and rodriguez the theorems of radford majid and bespalov in the case of hopf algebras with projection and the ones obtained by alonso and gonzalez for weak hopf algebras living in a symmetric category with split idempotents for example the weak theorem of blattner cohen and montgomery for weak hopf algebras with coalgebra splitting is one of them

c algebraic partial compact quantum groups abstract in this paper we introduce c algebraic partial compact quantum groups which are quantizations of topological groupoids with discrete object set and compact morphism spaces these c algebraic partial compact quantum groups are generalizations of hayashi s compact face algebras to the case where the object set can be infinite they form the c algebraic counterpart of an algebraic theory of partial compact quantum groups developed in an earlier paper by the author and t timmermann the correspondence between which will be dealt with in a separate paper as an interesting example to illustrate the theory we show how the dynamical quantum su 2 group as studied by etingof varchenko and koelink rosengren fits into this framework

semistable principal g bundles in positive characteristic let x be a normal projective variety defined over an algebraically closed field k of positive characteristic let g be a connected reductive group defined over k we prove that some frobenius pullback of a principal g bundle admits the canonical reduction ep such that its extension by p p rup is strongly semistable see theorem 5 1 then we show that there is only a small difference between semistability of a principal g bundle and semistability of its frobenius pullback see theorem 6 3 this and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable rational principal g bundles

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

poincare families of g bundles on a curve let g be a reductive group over an algebraically closed field k consider the moduli space of stable principal g bundles on a smooth projective curve c over k we give necessary and sufficient conditions for the existence of poincare bundles over open subsets of this moduli space and compute the orders of the corresponding obstruction classes this generalizes the previous results of newstead ramanan and balaji biswas nagaraj newstead to all reductive groups to all topological types of bundles and also to all characteristics

cocharacter closure and the rational hilbert mumford theorem for a field k let g be a reductive k group and v an affine k variety on which g acts using the notion of cocharacter closed g k orbits in v we prove a rational version of the celebrated hilbert mumford theorem from geometric invariant theory we initiate a study of applications stemming from this rationality tool a number of examples are discussed to illustrate the concept of cocharacter closure and to highlight how it differs from the usual zariski closure

effects of surface wedge corner and mixed boundary conditions on the local critical behaviour critical exponents for the o n spin model local field near a surface an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or comer is formed by mutually perpendicular d 1 dimensional walls each having von neumann special transition or dirichlet ordinary transitions boundary conditions the combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order e e 4 d the case of two or more von neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities

actions and coactions of finite quantum groupoids on von neumann algebras extensions of the matched pair procedure abstract in this work actions and coactions of finite c quantum groupoids are studied in an operator algebras context in particular we prove a double crossed product theorem and the existence of an universal von neumann algebra on which any finite groupoid acts outerly we give two actually different extensions of the matched pairs procedure in previous works n andruskiewitsch and s natale define for any matched pair of groupoids two c quantum groupoids in duality we give here an interpretation of them in terms of crossed products of groupoids using a single multiplicative partial isometry which gives a complete description of these structures the second extension deals only with groups to define an other type of finite c quantum groupoids

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

weak hopf bialgebras in this chapter the r bialgebroids and hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra r carries a so called separable frobenius structure separable frobenius structures on some algebra r are shown to correspond to separable frobenius structures on the forgetful functor from the category of r bimodules to the category of vector spaces based on that the bijection of chap 5 is refined to bijections between three structures for any algebra a first monoidal structures on the category of a modules together with separable frobenius structures on the forgetful functor to the category of vector spaces second bialgebroid structures on a over some base algebra r together with separable frobenius structures on r finally weak bialgebra structures on a the r bialgebroid a is a hopf algebroid if and only if the corresponding weak bialgebra a is a weak hopf algebra

an approach to quasi hopf algebras via frobenius coordinates abstract we study quasi hopf algebras and their subobjects over certain commutative rings from the point of view of frobenius algebras we introduce a type of radford formula involving an anti automorphism and the nakayama automorphism of a frobenius algebra then view several results in quantum algebras from this vantage point in addition separability and strong separability of quasi hopf algebras are studied as frobenius algebras

diagonal crossed products and l r smash products over weak hopf algebras the notions of the diagonal crossed product and the l r smash product over weak hopf algebras which can be seen as generalizations of smash products over weak hopf algebras are introduced and studied in particular the weak hopf algebraic structure of the drinfeld double of a finite dimensional weak hopf algebra which is a special case of diagonal crossed products is given moreover two canonical examples the kadison products and the connes moscovici products over weak hopf algebras are introduced and investigated

critical behaviour at an edge the critical behaviour of a magnetic system with o n spin symmetry bounded by two d 1 dimensional hyperplanes meeting at an angle alpha is studied within mean field theory and in d 4 epsilon dimensions new exponents emerge for correlation functions and magnetisation and susceptibilities for spins close to the edge they can be expressed in terms of known bulk and surface exponents together with a single new edge exponent which depends however on the angle alpha this exponent is computed to first order in epsilon

the spectra of quantum chains with free boundary conditions and virasoro algebras at the critical point the spectra of quantum chains with periodic and twisted boundary conditions are described by irreducible representations of two virasoro algebras with the same central charge the authors show that in the case of free boundary conditions the spectra can be understood in terms of irreducible representations of a single virasoro algebra for the ising and the three state potts models the corresponding irreducible representations are identified

surface order parameter in three dimensional percolation the authors calculate numerically by a transfer matrix algorithm the critical exponent of the surface order parameter s for the case of the ordinary transition their result s 0 98 0 02 agrees with recently published series expansion results s 1 04 0 05 and e expansion results with pade approximants s 1 01 0 06

logarithmic corrections for the percolative properties of the four state potts model clusters in the potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state droplets can be obtained by adding bonds with probability p 1 exp k between the sites in a cluster where k is the potts model s inverse temperature when q to 4 renormalization group rg fixed points describing clusters and droplets will coalesce leading to logarithmic corrections we calculate the precise form of these corrections used in a differential rg method our predictions are then tested using extensive monte carlo calculations theory and the simulations are found to be in excellent agreement

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

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

on modules associated to coalgebra galois extensions abstract for a given entwining structure a c involving an algebraa a coalgebrac and an entwining map c a a c a categorymca of right a c modules is defined and its structure analysed in particular the notion of a measuring of a c to a c is introduced and certain functors betweenmca andmca induced by such a measuring are defined it is shown that these functors are inverse equivalences iff they are exact or one of them faithfully exact and the measuring satisfies a certain galois type condition next left moduleseand right modules associated to ac galois extensionaofbare defined these can be thought of as objects dual to fibre bundles with coalgebracin the place of a structure group and a fibrev cross sections of such associated modules are defined as module mapse bor b it is shown that they can be identified with suitably equivariant maps from the fibre toa also it is shown that ac galois extension is cleft if and only ifa b cas leftb modules and rightc comodules the relationship between the moduleseand is studied in the case whenvis finite dimensional and in the case when the canonical entwining map is bijective

weak hopf algebras with projection and weak smash bialgebra structures abstract in this paper we study weak hopf algebras with projection if f h b g b h are morphisms of weak hopf algebras such that g f idh we prove that it is possible to find an object bh in the new category of weak yetter drinfeld modules that verifies similar conditions to the ones include in the definition of weak hopf algebra finally we define weak smash bialgebra structures and prove that under central and cocentral conditions bh and h determine an example of them

conformal invariance finite size scaling and surface magnetic exponent of the potts model in two dimensions the analysis of renormalisation group equations for the q state potts model in two dimensions is generalised to include the presence of a surface field hs besides the bulk one hb combined with conformal invariance results this allows prediction of a surface susceptibility behaviour delta 2 f delta hs delta hb approximately l 1 8 1 c lnl 15 16 for a q 4 system of size l at the critical temperature this prediction is checked by a monte carlo based finite size scaling analysis which also nicely reproduces the exact and conjectured magnetic surface exponents for q 2 and 3 respectively for the baxter wu model similar methods give results consistent with the q 4 behaviour without logarithmic corrections c 0

weak hopf algebras corresponding to uq sln we investigate the weak hopf algebras of li based on uq sln and sweedler s finite dimensional example we give weak hopf algebra isomorphisms between the weak generalizations of uq sln which are upgraded automorphisms of uq sln and hence give a classification of these structures as weak hopf algebras we also show how to decompose these examples into a direct sum which leads to unexpected isomorphisms between their algebraic structure

percolation in an edge percolation in a three dimensional lattice bounded by two free plane surfaces meeting at an angle chi is studied within a real space renormalisation scheme the edge critical behaviour at various surface and bulk transitions is analysed the author does not find evidence for a first order edge transition at higher values of alpha as has recently been observed for an ising model at an edge dependence of the edge scaling power ye on the opening angle alpha has been confirmed

weak c cleft extensions and weak galois extensions abstract for a weak entwining structure a c we formulate the notion of weak c galois extension with normal basis and we show that these galois extensions are equivalent to the weak c cleft extensions introduced in j n alonso alvarez j m fernandez vilaboa r gonzalez rodriguez a b rodriguez raposo weak c cleft extensions weak entwining structures and weak hopf algebras j algebra 284 2005 679 704

weak hopf algebras and weak yang baxter operators abstract in this paper we introduce the notions of weak yang baxter operator and weak braided hopf algebra we prove that it is possible to obtain examples of these notions working with yetter drinfeld modules associated to a weak hopf algebra h with invertible antipode finally we complete the study of the structure of weak hopf algebras with a projection obtaining a categorical equivalence between the category of weak hopf algebra projections associated to h and the category of hopf algebras in the non strict braided monoidal category of left left yetter drinfeld modules over h

some moduli stacks of symplectic bundles on a curve are rational abstract let c be a smooth projective curve of genus g 2 over a field k given a line bundle l on c let s ympl 2 n l be the moduli stack of vector bundles e of rank 2n on c endowed with a nowhere degenerate symplectic form b e e l up to scalars we prove that this stack is birational to b g m a s for some s if deg e n deg l is odd and c admits a rational point p c k as well as a line bundle of degree 0 with 2 o c it follows that the corresponding coarse moduli scheme of ramanathan stable symplectic bundles is rational in this case

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

a class of c algebraic locally compact quantum groupoids part ii main theory abstract this is part ii in our multi part series of papers developing the theory of a subclass of locally compact quantum groupoids quantum groupoids of separable type based on the purely algebraic notion of weak multiplier hopf algebras the definition was given in part i the existence of a certain canonical idempotent element e plays a central role in this part ii we develop the main theory discussing the structure of our quantum groupoids we will construct from the defining axioms the right left regular representations and the antipode map

morita theory for corings and cleft entwining structures abstract using the theory of corings we generalize and unify morita contexts introduced by chase and sweedler hopf algebras and galois theory in lecture notes in math vol 97 springer verlag berlin 1969 doi generalized smash products and morita contexts for arbitrary hopf algebras in j bergen s montgomery eds advances in hopf algebras in lecture notes in pure and appl math vol 158 dekker new york 1994 and cohen fischman and montgomery j algebra 133 1990 351 we discuss when the contexts are strict we apply our theory to corings arising from entwining structures and this leads us to the notion of cleft entwining structure

surface fractal dimension of two dimensional percolation the author determines the surface fractal dimension ds for two dimensional percolation from monte carlo data in finite systems the results agree with the prediction from conformal invariance ds 2 3

hopf algebroids with bijective antipodes axioms integrals and duals abstract motivated by the study of depth 2 frobenius extensions we introduce a new notion of hopf algebroid it is a 2 sided bialgebroid with a bijective antipode which connects the two left and right handed structures while all the interesting examples of the hopf algebroid of j h lu turn out to be hopf algebroids in the sense of this paper there exist simple examples showing that our definition is not a special case of lu s our hopf algebroids however belong to the class of l hopf algebras proposed by p schauenburg after discussing the axioms and some examples we study the theory of non degenerate integrals in order to obtain duals of hopf algebroids

classifying weak coideal subalgebras of weak hopf c algebras abstract we develop a general approach to the problem of classification of weak coideal c subalgebras of weak hopf c algebras as an example we consider weak hopf c algebras and their weak coideal c subalgebras associated with tambara yamagami categories

moduli space of principal sheaves over projective varieties let g be a connected reductive group the late ramanathan gave a notion of semi stable principal g bundle on a riemann surface and constructed a projective moduli space of such objects we generalize ramanathan fs notion and construction to higher dimension allowing also objects which we call semistable principal g sheaves in order to obtain a projective moduli space a principal g sheaf on a projective variety x is a triple p e o where e is a torsion free sheaf on x p is a principal g bundle on the open set u where e is locally free and o is an isomorphism between e u and the vector bundle associated to p by the adjoint representation we say it is semi stable if all filtrations e of e as sheaf of killing orthogonal algebras i e filtrations with e ui e i 1 and ei ej e e e i j have pei rke pe rkei 0 where pei is the hilbert polynomial of ei after fixing the chern classes of e and of the line bundles associated to the principal bundle p and characters of g we obtain a projective moduli space of semistable principal g sheaves we prove that in case dimx 1 our notion of semi stability is equivalent to ramanathan fs notion

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

the fundamental theorem for weak braided bimonads abstract the theories of hopf bialgebras and weak hopf bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product as first generalisations these notions were formulated for monoidal categories with braidings if needed the present authors developed a theory of bimonads and hopf monads h on arbitrary categories a employing distributive laws allowing for a general form of the fundamental theorem for hopf algebras for bimonads h properties of braided hopf bialgebras were captured by requiring a yang baxter operator h h h h the purpose of this paper is to extend the features of weak hopf bialgebras to this general setting including an appropriate form of the fundamental theorem this subsumes the theory of braided hopf algebras based on weak yang baxter operators as considered by alonso alvarez and others

on a morita equivalence between the duals of quantum su 2 and quantum e 2 abstract let su q 2 and e q 2 be woronowicz s q deformations of respectively the compact lie group su 2 and the non trivial double cover of the lie group e 2 of euclidean transformations of the plane we prove that in some sense their duals are morita equivalent locally compact quantum groups in more concrete terms we prove that the von neumann algebraic quantum groups l su q 2 and l e q 2 are unitary cocycle deformations of each other

finitely semisimple spherical categories and modular categories are self dual abstract we show that every essentially small finitely semisimple k linear additive spherical category for which k end 1 is a field is equivalent to its dual over the long canonical forgetful functor this includes the special case of modular categories in order to prove this result we show that the universal coend of the spherical category with respect to the long forgetful functor is self dual as a weak hopf algebra

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

surface and edge exponents for the spreading of 3d percolation monte carlo results are presented for the spreading of 3d percolation where the seed consists of an infinite straight line this line can either be in the bulk of the material into which spreading is possible on a planar surface of the material or on a rectangular edge in the last two cases spreading occurs only into angular regions with 180 degrees resp 90 degrees while the mean distance grows at p pc in all three cases with the same exponent the average number of growth sites grows resp vanishes in all three cases with different exponents in particular this implies that at p pc the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension 1

theory of boundary effects in invasion percolation we study the boundary effects in invasion percolation ip with and without trapping we find that the presence of boundaries introduces a new set of surface critical exponents as in the case of standard percolation numerical simulations show a fractal dimension for the region of the percolating cluster near the boundary remarkably different from the bulk one in fact on the surface we find a value of for ip with trapping compared with the bulk value of we find a logarithmic crossover from surface to bulk fractal properties as one would expect from the finite size theory of critical systems the distribution of the quenched variables on the growing interface near the boundary self organizes into an asymptotic shape characterized by a discontinuity at a value which coincides with the bulk critical threshold the exponent of the boundary avalanche distribution for ip without trapping is this value is very near to the bulk one then we conclude that only the geometrical properties fractal dimension of the model are affected by the presence of a boundary while other statistical and dynamical properties are unchanged furthermore we are able to present a theoretical computation of the relevant critical exponents near the boundary this analysis combines two recently introduced theoretical tools the fixed scale transformation and the run time statistics which are particularly suited for the study of irreversible self organized growth models with quenched disorder our theoretical results are in rather good agreement with numerical data

semisimple weak hopf algebras abstract we develop the theory of semisimple weak hopf algebras and obtain analogues of a number of classical results for ordinary semisimple hopf algebras we prove a criterion for semisimplicity and analyze the square of the antipode s 2 of a semisimple weak hopf algebra a we explain how the frobenius perron dimensions of irreducible a modules and eigenvalues of s 2 can be computed using the inclusion matrix associated to a a trace formula of larson and radford is extended to a relation between the categorical and frobenius perron dimensions of a finally an analogue of the class equation of kac and zhu is established and properties of a module algebras and their dimensions are studied

ising model at an edge a position space renormalisation group approach the critical behaviour of a three dimensional ising model bounded by two plane surfaces meeting at an angle alpha is studied using the migdal kadanoff and two terminal cluster renormalisation methods the authors obtain expressions for the edge fixed point and for the independent edge exponent yhe at the various transitions they find that yhe is not completely universal but depends on the angle alpha and that it shows some unexpected features

bulk surface and hull fractal dimension of critical ising clusters in d 2 the authors present accurate numerical calculations of the fractal dimension d and surface dimension ds of the critical ising cluster in d 2 the results clearly support the values d 187 96 ds 5 6 which are consistent with ising clusters being described by tricritical q 1 potts model exponents from this the hull dimension dh of critical ising clusters is found to be dh 11 8 consistent with numerical work of other authors

principal bundles on projective varieties and the donaldson uhlenbeck compactification let h be a semisimple algebraic group we prove the semistable reduction theorem for semistable principal h bundles over a smooth projective variety x defined over the field c when x is a smooth projective surface and h is simple we construct the algebro geometric donaldson uhlenbeck compactification of the moduli space of semistable principal h bundles with fixed characteristic classes and describe its points for large characteristic classes we show that the moduli space of stable principal h bundles is non empty

conformal invariance and surface critical behaviour of a quantum chain with three spin interaction an ising model with three spin interaction is investigated by finite size scaling applying free boundary conditions the surface critical properties of the model were determined by conformal invariance taking into account logarithmic corrections the surface critical exponents obtained are in accordance with the four state potts values

crossed products for weak hopf algebras with coalgebra splitting in this paper we prove that if g b h is a morphism of weak hopf algebras which is split as a coalgebra morphism then there exists a subalgebra bh of b morphisms bh h bh bh bh h h bh and an isomorphism of algebras and right h comodules bb b bh h being bh h a subobject of bh h with its algebra structure defined by a crossed product also we obtain the dual results and as a consequence we prove the radford s theorem for weak hopf algebras

moduli spaces for principal bundles in arbitrary characteristic in this article we solve the problem of constructing moduli spaces of semistable principal bundles and singular versions of them over smooth projective varieties over algebraically closed ground fields of positive characteristic

weak hopf algebras ii representation theory dimensions and the markov trace abstract if a is a weak c hopf algebra then the category of finite dimensional unitary representations of a is a monoidal c category with its monoidal unit being the gns representation d e associated to the counit e this category has isomorphic left dual and right dual objects which leads as usual to the notion of a dimension function however if e is not pure the dimension function is matrix valued with rows and columns labeled by the irreducibles contained in d e this happens precisely when the inclusions a l a and a r a are not connected still there exists a trace on a which is the markov trace for both inclusions we derive two numerical invariants for each c wha of trivial hypercenter these are the common indices i and of the haar respectively markov conditional expectations of either one of the inclusions a l r a or a l r a in generic cases i in the special case of weak kac algebras we reproduce d nikshych s result 2000 j operator theory to appear by showing that i and is always an integer