deformed algebras from elliptic sl n algebras we extend to the case the results that we previously obtained on the construction of algebras from the elliptic algebra the elliptic algebra at the critical level c n has an extended center containing trace like operators t z families of poisson structures indexed by n n 1 2 integers defining q deformations of the algebra are constructed the operators t z also close an exchange algebra when for it becomes abelian when in addition p qnh where h is a non zero integer the poisson structures obtained in these classical limits contain different q deformed algebras depending on the parity of h characterizing the exchange structures at p qnh as new algebras

sun wong type theorems for second order damped elliptic equations abstract the oscillation criteria of sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 are extended to the forced second order damped elliptic differential equation with mixed nonlinearities i j 1 n d i a i j x d j y i 1 n b i x d i y c x y e x when n 1 our results include and improve those of nasr a h nasr sufficient conditions for the forced super linear second order differential equations with oscillatory potential proc amer math soc 126 1998 123 125 for a forced second order superlinear differential equation wong j s w wong oscillation criteria for a forced second order linear differential equations j math anal appl 231 1999 235 240 for a forced second order linear differential equation sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 for a forced second order differential equation with mixed nonlinearities when n 2 our results are more general and improve the oscillation criteria of zhuang r k zhuang annual oscillation criteria for second order nonlinear elliptic differential equations j comput anal math 217 2008 268 276 for a forced second order undamped elliptic differential equation

the drinfeld realization of the elliptic quantum group bq λ a2 2 we construct a realization of the l operator satisfying the rll relation of the face type elliptic quantum group bq a2 2 the construction is based on the elliptic analog of the drinfeld currents of uq a2 2 which forms the elliptic algebra uq p a2 2 we give a realization of the elliptic currents e z f z and k z as a tensor product of the drinfeld currents of uq a2 2 and a heisenberg algebra in the level one representation we also give a free field realization of the elliptic currents applying these results we derive a free field realization of the uq p a2 2 analog of the bq a2 2 intertwining operators the resultant operators coincide with those of the vertex operators in the dilute al model which is known to be a rsos restriction of the a2 2 face model

pullback and pushout constructions in c algebra theory abstract a systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of c algebras particular emphasis is given to the relations with tensor products both with the minimal and the maximal c tensor norm thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group general tensor products between diagrams are also investigated the relations between the theory of extensions and pullback and pushout diagrams are explored in some detail the crowning result is that if three short exact sequences of c algebras are given with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals algebras and quotients then the three new c algebras will again form a short exact sequence under some mild extra conditions as a generalization of a theorem of t a loring it is shown that each morphism between a pair of c algebras combined with its extension to the stabilized algebras gives rise to a pushout diagram this result has applications to corona extendibility and conditional projectivity finally the pullback and pushout constructions are applied to the class of noncommutative cw complexes defined by s eilers t a loring and g k pedersen j reine angew math 1998 499 101 143 to show that this category is stable under tensor products and under restricted direct sums

an elliptic algebra and the fusion rsos model we introduce an elliptic algebra with and present its free boson representation at generic level k we show that this algebra governs a structure of the space of states in the k fusion rsos model specified by a pair of positive integers r k or equivalently a q deformation of the coset conformal field theory extending the work by lukyanov and pugai corresponding to the case k 1 we give a full set of screening operators for k 1 the algebra has two interesting degeneration limits p 0 and p 1 the former limit yields the quantum affine algebra whereas the latter yields the algebra the scaling limit of the elliptic algebra using this correspondence we also obtain the highest component of two types of vertex operators which can be regarded as q deformations of the primary fields in the coset conformal field theory

dynamical centers for the elliptic quantum algebra we identify generating functionals that satisfy dynamical exchange relations with the lax matrices defining the face type elliptic quantum algebra when the central charge takes the two possible values these generating functionals are constructed as quadratic trace like objects in terms of the lax matrices the obtained structures are characterized as dynamical centers i e the centrality property is deformed by dynamical shifts for these values the functionals define genuine abelian subalgebras of

a class of partition functions associated with e_ tau eta gl_3 by izergin korepin analysis recently a class of partition functions associated with higher rank rational and trigonometric integrable models were introduced by foda and manabe we use the dynamical r matrix of the elliptic quantum group e gl3 to introduce an elliptic analog of the partition functions associated with e gl3 we investigate the partition functions of foda manabe type by developing a nested version of the elliptic izergin korepin analysis and present the explicit forms as symmetrization of multivariable elliptic functions we show that special cases are essentially the elliptic weight functions introduced in the works by rimanyi tarasov and varchenko konno and felder rimanyi and varchenko

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities abstract we present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities p t x q t x i 1 n q i t x i sgn x e t where p t q t q i t e t c 0 p t is positive and differentiable 1 m 1 m 1 n no restriction is imposed on the forcing term e t to be the second derivative of an oscillatory function when n 1 our results reduce to those of el sayed m a el sayed an oscillation criterion for a forced second order linear differential equation proc amer math soc 118 1993 813 817 wong j s w wong oscillation criteria for a forced second linear differential equations j math anal appl 231 1999 235 240 sun ou and wong y g sun c h ou j s w wong interval oscillation theorems for a linear second order differential equation comput math appl 48 2004 1693 1699 for the linear equation nazr a h nazr sufficient conditions for the oscillation of forced super linear second order differential equations with oscillatory potential proc amer math soc 126 1998 123 125 for the superlinear equation and sun and wong y g sun j s w wong note on forced oscillation of nth order sublinear differential equations j math anal appl 298 2004 114 119 for the sublinear equation

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 haagerup property for locally compact quantum groups the haagerup property for locally compact groups is generalised to the context of locally compact quantum groups with several equivalent characterisations in terms of the unitary representations and positive definite functions established in particular it is shown that a locally compact quantum group g has the haagerup property if and only if its mixing representations are dense in the space of all unitary representations for discrete g we characterise the haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group hat g by the existence of a real proper cocycle on g and further if g is also unimodular we show that the haagerup property is a von neumann property of g this extends results of akemann walter bekka cherix valette and jolissaint to the quantum setting and provides a connection to the recent work of brannan we use these characterisations to show that the haagerup property is preserved under free products of discrete quantum groups

the elliptic algebra and the drinfeld realization of the elliptic quantum group by using the elliptic analogue of the drinfeld currents in the elliptic algebra we construct a l operator which satisfies the rll relations characterizing the face type elliptic quantum group for this purpose we introduce a set of new currents k j v 1 leq j leq n in as in the n 2 case we find a structure of as a certain tensor product of and a heisenberg algebra in the level one representation we give a free field realization of the currents in using the coalgebra structure of and the above tensor structure we derive a free field realization of the analogue of intertwining operators the resultant operators coincide with those of the vertex operators in the a n 1 1 type face model

orthogonal free quantum group factors are strongly 1 bounded abstract we prove that the orthogonal free quantum group factors l f o n are strongly 1 bounded in the sense of jung in particular they are not isomorphic to free group factors this result is obtained by establishing a spectral regularity result for the edge reversing operator on the quantum cayley tree associated to f o n and combining this result with a recent free entropy dimension rank theorem of jung and shlyakhtenko

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

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

the elliptic quantum algebra u_ q hbox scriptsize p widehat sl_n and its vertex operators we construct a realization of the elliptic quantum algebra for any given level k in terms of free boson fields and their twisted partners it can be considered as the elliptic deformation of the wakimoto realization of the quantum affine algebra we also construct a family of screening currents which commute with the currents of up to total q differences and we give explicit twisted expressions for the type i and type ii vertex operators of by twisting the known results of the type i vertex operators of the quantum affine algebra and the new results of the type ii vertex operators of we obtained in this paper

integral averages and oscillation criteria for half linear partial differential equation the technique of weighted integral averages known in the oscillation theory of ordinary differential equations is extended to the half linear partial differential equation e div email protected email protected p email protected u c x u p 2u 0 p 1 this technique is used to obtain new oscillation criteria for e on the unbounded domains

integration over quantum permutation groups abstract we find a combinatorial formula for the haar measure of quantum permutation groups this leads to a dynamic formula for laws of diagonal coefficients explaining the poisson free poisson convergence result for characters

oscillation of the half linear pde in general exterior domains the variational approach abstract in the paper a variational technique is used to show how to derive new oscillation criteria for the half linear partial differential equation via the oscillation criteria for the ordinary half linear differential equation in contrast to the results usually found in the literature general types of unbounded domains can be considered

the free entropy dimension of hyperfinite von neumann algebras suppose m is a hyperfinite von neumann algebra with a normal tracial state and 1 n is a set of selfadjoint generators for m we calculate 0 1 n the modified free entropy dimension of 1 n moreover we show that 0 1 n depends only on m and consequently 0 1 n is independent of the choice of generators for m in the course of the argument we show that if b 1 b n is a set of selfadjoint generators for a von neumann algebra r with a normal tracial state and b 1 b n has finite dimensional approximants then 0 n 0 b 1 b n for any hyperfinite von neumann subalgebra n of r combined with a result by voiculescu this implies that if r has a regular diffuse hyperfinite von neumann subalgebra then 0 b 1 b n 1

the connes embedding property for quantum group von neumann algebras for a compact quantum group g of kac type we study the existence of a haar trace preserving embedding of the von neumann algebra l g into an ultrapower of the hyperfinite ii 1 factor the connes embedding property for l g we establish a connection between the connes embedding property for l g and the structure of certain quantum subgroups of g and use this to prove that the ii 1 factors l o n and l u n associated to the free orthogonal and free unitary quantum groups have the connes embedding property for all n 4 as an application we deduce that the free entropy dimension of the standard generators of l o n equals 1 for all n 4 we also mention an application of our work to the problem of classifying the quantum subgroups of o n

towards a cladistics of double yangians and elliptic algebras a self contained description of algebraic structures obtained by combinations of various limit procedures applied to vertex and face sl 2 elliptic quantum affine algebras is given new double yangian structures of dynamical type are defined connections between these structures are established a number of them take the form of twist like actions these are conjectured to be evaluations of universal twists

examples of weakly amenable discrete quantum groups abstract we prove that the free orthogonal and free unitary quantum groups f o n and f u n are weakly amenable and that their cowling haagerup constant is equal to 1 this is achieved by estimating the completely bounded norm of the projections on the coefficients of irreducible representations of their compact duals an argument of monoidal equivalence then allows us to extend this result to quantum automorphism groups of finite spaces and even yields some examples of weakly amenable non unimodular discrete quantum groups with the haagerup property

parametrization of semi dynamical quantum reflection algebra we construct sets of structure matrices for the semi dynamical reflection algebra solving the yang baxter type consistency equations extended by the action of an automorphism of the auxiliary space these solutions are parametrized by dynamical conjugation matrices drinfel d twist representations and quantum non dynamical r matrices they yield factorized forms for the monodromy matrices

residually finite quantum group algebras abstract we show that provided n 3 the involutive hopf algebra a u n coacting universally on an n dimensional hilbert space has enough finite dimensional representations in the sense that every non zero element acts non trivially in some finite dimensional representation this implies that the discrete quantum group with group algebra a u n is maximal almost periodic i e it embeds in its quantum bohr compactification answering a question posed by p soltan in 21 we also prove analogous results for the involutive hopf algebra b u n coacting universally on an n dimensional hilbert space equipped with a non degenerate bilinear form

ordinary differential equations in the oscillation theory of partial half linear differential equation abstract in the paper we study the damped half linear partial differential equation div a x u p 2 u b x u p 2 u c x u p 2 u 0 using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations using careful radialization we improve several known oscillation criteria

symmetric functions and wavefunctions of xxz type six vertex models and elliptic felderhof models by izergin korepin analysis we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an elliptic extension of the trigonometric felderhof model we show that the wavefunctions can be expressed as one parameter deformations of an elliptic analog of the vandermonde determinant and elliptic symmetric functions we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an

free field approach to the dilute al models we construct a free field realization of vertex operators of the dilute al models along with the felder complex for l 3 we also study an e8 structure in terms of the deformed virasoro currents

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

large deviation bounds for matrix brownian motion we prove large deviation bounds for the convergence of hermitian matrix valued brownian motion towards free brownian motion as a consequence we obtain upper and lower bounds on the microstates entropy introduced by voiculescu 24

conjugacy criteria for half linear ode in the theory of pde with generalized p laplacian and mixed powers abstract in this paper we use conjugacy criteria for a half linear ordinary differential equation to prove some nonexistence results for a partial differential equation with generalized p laplacian and mixed derivatives div a x y p 2 y b x y p 2 y c x y p 2 y i 1 m c i x y p i 2 y e x the results can be used also to study equations with strongly non radial potential c x since the usual dependence on the mean value of c x over a sphere can be replaced by a more general term

free field construction for the abf models in regime ii the wakimoto construction for the quantum affine algebra u q widehat mathfrak s mathfrak l 2 admits a reduction to the q deformed parafermion algebras we interpret the latter theory as a free field realization of the andrews baxter forrester models in regime ii we give multi particle form factors of some local operators on the lattice and compute their scaling limit where the models are described by a massive field theory with mathbb z k symmetric minimal scattering matrices

the elliptic algebra and the deformation of wn algebra after reviewing the recent results on the drinfeld realization of the face type elliptic quantum group b q sl n by the elliptic algebra u q p sl n we investigate a fusion of the vertex operators of u q p sl n the basic generating functions j z 1 j n 1 of the deformed w n algebra are derived explicitly

topological generation and matrix models for quantum reflection groups abstract we prove that the von neumann algebras of quantum permutation groups and quantum reflection groups have the connes embedding property we do this by establishing several new topological generation results for the quantum permutation groups s n and the quantum reflection groups h n s we use these results to show that these quantum groups admit sufficiently many matrix models in particular all of these quantum groups have residually finite discrete duals and are in particular hyperlinear and certain flat matrix models for s n are inner faithful

paths in quantum cayley trees and l 2 cohomology abstract we study existence uniqueness and triviality of path cocycles in the quantum cayley graph of universal discrete quantum groups in the orthogonal case we find that the unique path cocycle is trivial in contrast with the case of free groups where it is proper in the unitary case it is neither bounded nor proper from this geometrical result we deduce the vanishing of the first l 2 betti number of a o i n

on the vertex operators of the elliptic quantum algebra uq p sl2 k a realization of the elliptic quantum algebra uq p sl2 for any given level k is constructed in terms of three free boson fields and their accompanying twisted partners it can be viewed as the elliptic deformation of the wakimoto realization two screening currents are constructed they commute or anticommute with uq p sl2 modulo total q differences the free field realization for two types of vertex operators nominated as the type i and the type ii vertex operators is presented the twisted version of the two types of vertex operators are also obtained they all play crucial roles in calculating correlation functions

bootstrap equations and correlation functions for the heisenberg xyz antiferromagnet we present two kinds of integral solutions to the quantum knizhnik zamolodchikov equations for the 2n point correlation functions of the heisenberg xyz antiferromagnet our first integral solution can be obtained from those for the cyclic sos model by using the vertex face correspondence by the construction the sum with respect to the local height variables k0 k1 k2n of the cyclic sos model remains other than the n fold integral in the first solution in order to perform these summations we improve this to find the second integral solution of r 1 n fold integral for r 1 where r is a parameter of the xyz model furthermore we discuss the relations among our formula lashkevich pugai s formula and that of shiraishi