quantized knizhnik zamolodchikov equations quantum yang baxter equation and difference equations for q hypergeometric functions thesl2 quantized knizhnik zamolodchikov equations are solved inq hypergeometric functions new difference equations are derived for generalq hypergeometric functions the equations are given in terms of quantum yang baxter matrices and have the form similar to quantum knizhnik zamolodchikov equations for quantum affine algebras introduced by frenkel and reshetikhin

exchange dynamical quantum groups for any simple lie algebra and any complex number q which is not zero or a nontrivial root of unity but may be equal to 1 we construct a dynamical quantum group hopf algebroid whose representation theory is essentially the same as the representation theory of the quantum group uq this dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of uq and is an algebraic structure standing behind these relations

pinning of a rough interface by an external potential abstract an analysis is given of the behavior of an interface between two phases in the presence of an external pinning potential in the solid on solid limit of the two dimensional ising model it is found that the potential turns a rough interface into a smooth one except in the case of a boundary potential where a minimum potential strength is required the connection with the roughening transition found by abraham is discussed the interface width is calculated as a function of the potential parameters in the limit of a weak pining potential

universal vertex irf transformation for quantum affine algebras we construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras which is an extension of our previous work to uq ar 1 this universal solution has a simple gauss decomposition which is constructed using sevostyanov s characters of twisted quantum borel algebras we show that in the evaluation representations it gives a vertex face transformation between a vertex type solution and a face type solution of the quantum dynamical yang baxter equation in particular in the evaluation representation of uq a1 1 it gives baxter s well known transformation between the 8 vertex model and the interaction round faces irf height model

short distance expansion for the local susceptibility and pair correlation function at continuous wetting transitions a scaling ansatz is proposed for the density profile rho z with z the distance normal from the substrate for a fluid undergoing a continuous wetting transition at a wall the ansatz predicts that in the fluctuation dominated regime the local susceptibility delta rho z delta mu t with mu the chemical potential has a simple power law position dependence for short distances z zeta perpendicular to the perpendicular correlation length the power law is determined by the wetting critical exponents which is different for complete and critical wetting the scaling predictions are confirmed in dimension d 2 by explicit analysis for an interfacial hamiltonian model of the wetting transition for critical wetting a scaling ansatz for the form of the transverse moments of the two point correlation function g is postulated which now yields the z1 z2 dependence of g in terms of critical exponents calculations again confirm the scaling theory for d 2 the analysis highlights the different qualitative and quantitative features of the response functions for critical and complete wetting for fluids with short ranged forces

triangular dynamical r matrices and quantization abstract we study some general aspects of triangular dynamical r matrices using poisson geometry we show that a triangular dynamical r matrix r h 2 g always gives rise to a regular poisson manifold using the fedosov method we prove that non degenerate triangular dynamical r matrices i e those such that the corresponding poisson manifolds are symplectic are quantizable and that the quantization is classified by the relative lie algebra cohomology h2 g h

identities between q hypergeometric and hypergeometric integrals of different dimensions abstract given complex numbers m1 l1 and nonnegative integers m2 l2 such that m1 m2 l1 l2 for any a b 0 min m2 l2 we define an l2 dimensional barnes type q hypergeometric integral ia b z m1 m2 l1 l2 and an l2 dimensional hypergeometric integral ja b z m1 m2 l1 l2 the integrals depend on complex parameters z and we show that ia b z m1 m2 l1 l2 equals ja b e z l1 l2 m1 m2 up to an explicit factor thus establishing an equality of l2 dimensional q hypergeometric and m2 dimensional hypergeometric integrals the identity is based on the gl k gl n duality for the qkz and dynamical difference equations

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

a universal parameterization of finite size effects for interfaces in two dimensions conformal mappings and local scale invariance the author shows that all equilibrium statistical properties of an interface confined in a strip geometry with arbitrary aspect ratio exactly at a second order fluctuation dominated interfacial unbinding wetting transition are determined by a single number q for interfacial binding potentials that are conformally mapped from the semi infinite plane the parameter q distinguishes the fluctuation regimes describing the wetting transition and can be directly related to wetting critical exponents in the strong fluctuation regime one finds q 0 whilst in the weak fluctuation regime q 1 the values of q at all intermediate fluctuation scaling regimes are also determined he shows that the eigenstates of the transfer differential operator for conformally mapped marginal long ranged potentials are the simplest possible generalization of the eigenstates corresponding to systems with short ranged forces he speculates that the universal parametrization of the finite size effects at fluctuation dominated wetting transitions is a consequence of local scale invariance

enriched p partitions and peak algebras abstract we develop a more general view of stembridge s enriched p partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group initially we focus on commutative peak algebras spanned by sums of permutations with the same number of peaks where we consider several variations on the definition of peak whereas stembridge s enriched p partitions are related to quasisymmetric functions the dual coalgebra of solomon s type a descent algebra our generalized enriched p partitions are related to type b quasisymmetric functions the dual coalgebra of solomon s type b descent algebra using these functions we move on to explore non commutative peak algebras spanned by sums of permutations with the same set of peaks while some of these algebras have been studied before our approach gives explicit structure constants with a combinatorial description

trees and spatial topology change in causal dynamical triangulations generalized causal dynamical triangulations generalized cdt is a model of two dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur we solve the model at the discretized level using bijections between quadrangulations and trees in the continuum limit scaling limit the amplitudes are shown to agree with known formulas and explicit expressions are obtained for loop propagators and two point functions it is shown that from a combinatorial point of view generalized cdt can be viewed as the scaling limit of planar maps with a finite number of faces and we determine the distance function on this ensemble of planar maps finally the relation with planar maps is used to illuminate the mysterious identity of certain continuum cylinder amplitudes

scaling and local scale invariance for wetting transitions and confined interfaces the author studies the scaling behaviour of the magnetization profile m z of an ising like magnet in a parallel plate geometry with opposite h1 h2 surface fields in the limit of large plate separation l and zero bulk field m z is a scaled function of z l for temperatures tw or t tc where tw is the critical wetting temperature of each semi infinite surface the exact form of m z depends on whether t tw tc t tw or t tc the influence of the far wall on the magnetization near one surface is long ranged and is determined for t tc by wetting critical exponents the author shows that the results of the capillary wave model for a local one point function probability distribution energy density may be derived by conformally mapping the corresponding quantity defined in the semi infinite geometry at the appropriate wetting transition the author discusses the nature of local scale invariance for wetting transitions and speculates as to why conformal invariance has application to local one point functions

form factors and action of on cycles let be a sequence of skew symmetric polynomials in x1 xl satisfying degxjpn l n 1 whose coefficients are symmetric laurent polynomials in z1 zn we call p an cycle if holds for all n l these objects arise in integral representations for form factors of massive integrable field theory i e the su 2 invariant thirring model and the sine gordon model the variables a logxa are the integration variables and j logzj are the rapidity variables to each cycle there corresponds a form factor of the above models conjecturally all form factors are obtained from the cycles in this paper we define an action of on the space of cycles there are two sectors of cycles depending on whether n is even or odd using this action we show that the character of the space of even resp odd cycles which are polynomials in z1 zn is equal to the level 1 irreducible character of with lowest weight 0 resp 1 we also suggest a possible tensor product structure of the full space of cycles

dynamical yang baxter equation and quantum vector bundles we develop a categorical approach to the dynamical yang baxter equation dybe for arbitrary hopf algebras in particular we introduce the notion of a dynamical extension of a monoidal category which provides a natural environment for quantum dynamical r matrices dynamical twists etc in this context we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits we build a dynamical twist for any pair of a reductive lie algebra and its levi subalgebra using this twist we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive lie groups

pairings and actions for dynamical quantum groups abstract dynamical quantum groups constructed from a frst construction using a solution of the quantum dynamical yang baxter equation are equipped with a natural pairing the interplay of the pairing with structures corepresentations and dynamical representations is studied and natural left and right actions are introduced explicit details for the elliptic u 2 dynamical quantum group are given and the pairing is calculated explicitly in terms of elliptic hypergeometric functions dynamical analogues of spherical and singular vectors for corepresentations are introduced

free field representation for massive integrable models a new approach to massive integrable models is considered it allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form factors in thesu 2 thirring and sine gordon models

tensor product of principal unitary representations of quantum lorentz group and askey wilson polynomials we study the tensor product of principal unitary representations of the quantum lorentz group prove a decomposition theorem and compute the associated intertwiners we show that these intertwiners can be expressed in terms of complex continuations of 6j symbols of uq su 2 these intertwiners are expressed in terms of q racah polynomials and askey wilson polynomials the orthogonality of these intertwiners imply some relation mixing these two families of polynomials the simplest of these relations is the orthogonality of askey wilson polynomials

jackson integrals of jordan pochhammer type and quantum knizhnik zamolodchikov equations we show that theq difference systems satisfied by jackson integrals of jordan pochhammer type give a class of the quantum knizhnik zamolodchikov equation for u q widehat mathfrak s mathfrak l 2 in the sense of frenkel and reshetikhin

a characteristic equation associated to an eigenvalue problem on c2 networks abstract a characteristic equation for the spectrum of an eigenvalue problem on c 2 networks is deduced and those special cases are investigated in which the spectrum depends mainly on the structure of the abstract graph belonging to the c 2 network

connection problems for quantum affine kz equations and integrable lattice models cherednik attached to an affine hecke algebra module a compatible system of difference equations called quantum affine knizhnik zamolodchikov kz equations in the case of a principal series module we construct a basis of power series solutions of the quantum affine kz equations relating the bases for different asymptotic sectors gives rise to a weyl group cocycle which we compute explicitly in terms of theta functions

quantum groupoids and dynamical categories abstract in this paper we realize the dynamical categories introduced in our previous paper as categories of modules over bialgebroids we study the bialgebroids arising in this way we define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to the dynamical categories we show that dynamical twists over an arbitrary base give rise to bialgebroid twists we prove that the classical dynamical r matrices over an arbitrary base manifold are in one to one correspondence with a special class of coboundary lie bialgebroids

explicit quantization of dynamical r matrices for finite dimensional semisimple lie algebras 1 1 classical r matrices in the early eighties belavin and drinfeld bd classified nonskewsymmetric classical r matrices for simple lie algebras it turned out that such r matrices up to isomorphism and twisting by elements from the exterior square of the cartan subalgebra are classified by combinatorial objects which are now called belavin drinfeld triples by definition a belavin drinfeld triple for a simple lie algebra g is a triple 1 2 t where 1 2 are subsets of the dynkin diagram of g and t 1 2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition if 1 then there exists k such that t k 1 1 but t 1 the r matrix corresponding to such a triple is given by a certain explicit formula these results generalize in a straightforward way to semisimple lie algebras in s the third author generalized the work of belavin and drinfeld and classified classical nonskewsymmetric dynamical r matrices for simple lie algebras it turns out that they have an even simpler classification up to gauge transformations they are classified by generalized belavin drinfeld triples which are defined as the usual belavin drinfeld triples but without any nilpotency condition the dynamical rmatrix corresponding to such a triple is given by a certain explicit formula as before these results can be generalized to semisimple lie algebras

quantization of classical dynamical r matrices with nonabelian base we construct some classes of dynamical r matrices over a nonabelian base and quantize some of them by constructing dynamical pseudo twists in the sense of xu this way we obtain quantizations of r matrices obtained in earlier work of the second author with schiffmann and varchenko a part of our construction may be viewed as a generalization of the donin mudrov nonabelian fusion construction we apply these results to the construction of equivariant star products on poisson homogeneous spaces which include some homogeneous spaces introduced by de concini

non ornstein zernike surface structure factor for complete wetting in three and above dimensions we have derived a closed form expression for the transverse fourier transform g 0 0 q or surface structure factor of the surface spin spin correlation function near a complete wetting transition from landau theory whilst g 0 0 q contains isolated singularities poles in the complex wavevector plane it does not have a simple omstein zernike oz form instead the function exhibits two limiting oz like behaviours characteristic of intrinsic and coherent capillary wave like fluctuations depending on the value of the scaling variable q mod h mod sub vco we also discuss the decay of surface correlations in real space and identify the appropriate singular long ranged contribution in contrast to the second moment correlation length the true correlation length at the wall diverges as mod h mod sub v infinity in the limit of complete wetting

matrix difference equations for the supersymmetric lie algebra sl 2 1 and the off shell bethe ansatz based on the rational r matrix of the supersymmetric sl 2 1 matrix difference equations are solved by means of a generalization of the nested algebraic bethe ansatz these solutions are shown to be of highest weight with respect to the underlying graded lie algebra structure

elliptic stable envelopes we construct stable envelopes in equivariant elliptic cohomology of nakajima quiver varieties in particular this gives an elliptic generalization of the results of arxiv 1211 1287 we apply them to the computation of the monodromy of q difference equations arising the enumerative k theory of rational curves in nakajima varieties including the quantum knizhnik zamolodchikov equations

correlations at a liquid gas interface asymptotic analysis for weak gravity density density correlation functions at the phase separating layer in a two dimensional solid on solid lattice model are studied the authors perform for weak gravitational fields an exact asymptotic analysis and obtain explicit expressions a recent numerical analysis by stecki and dudowicz 1986 is shown to be in good agreement with their exact expansion

trace functionals on noncommutative deformations of moduli spaces of flat connections abstract we describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a riemann surface the resulting algebra is a variant of the quantum moduli algebra introduced by alekseev grosse and schomerus and buffenoir and roche we construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of fedosov and nest and tsygan via verlinde s formula

the peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

isomorphism of two realizations of quantum affine algebra u_q widehat mathfrak g mathfrak l text n text we establish an explicit isomorphism between two realizations of the quantum affine algebra u q widehat mathfrak g mathfrak l text n text given previously by drinfeld and reshetikhin semenov tian shansky our result can be considered as an affine version of the isomorphism between the drinfield jimbo and the faddeev reshetikhin takhtajan constructions of the quantum algebra u q mathfrak g mathfrak l n

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

universal amplitude exponent relations for interfaces and conformal mappings the author considers the finite size fs scaling behaviour of the transverse correlation lengths xi sub n for an interface confined in a strip geometry exactly at a second order fluctuation dominated wetting transition he shows that the dimensionless amplitude ratios a sub n sigma l2 kbt xi sub n with sigma the surface stiffness coefficient are related to the short distance expansion critical exponent theta by a sub n pi 2n n theta 1 2 for all fluctuation dominated wetting transitions provided the binding potential v y in the capillary wave model is conformally mapped from the semi infinite plane he argues that the existence of this amplitude exponent relation reflects the conformal invariance of the capillary wave model lagrange density and lists a number of one point functions which can be shown to exhibit this invariance explicitly

on representations of the elliptic quantum group e τ η sl 2 we describe representation theory of the elliptic quantum groupe sl 2 it turns out that the representation theory is parallel to the representation theory of the yangiany sl 2 and the quantum loop group u q widetilde sl 2 we introduce basic notions of representation theory of the elliptic quantum groupe sl 2 and construct three families of modules evaluation modules cyclic modules one dimensional modules we show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one dimensional module we describe fusion of finite dimensional evaluation modules in particular we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients we describe the determinant element of the elliptic quantum group representation theory becomes special ifn m l wheren m l are integers we indicate some new features in this case

geometry and classificatin of solutions of the classical dynamical yang baxter equation the classical yang baxter equation cybe is an algebraic equation central in the theory of integrable systems its nondegenerate solutions were classified by belavin and drinfeld quantization of cybe led to the theory of quantum groups a geometric interpretation of cybe was given by drinfeld and gave rise to the theory of poisson lie groups

magnetization profiles for a ising strip with opposite surface fields we obtain an exact solution of the d 2 infinite ising strip with surface fields of opposite signs earlier predictions by parry et al are confirmed of the crucial importance of wetting for a system confined between two parallel walls that exert competitive surface fields the transition of the magnetization profile m z from the partial wetting regime below the critical wetting temperature at a single wall with a surface field to the soft mode single phase in the temperature range where is the critical temperature of the bulk system is described in detail a scaling ansatz is verified for the singular part of the surface excess free energy and for the profile the modification of the magnetization profile near one wall due to wetting is studied near the wall at the profile is found to be a linear function of the scaled distance but its slope is different than for the case of perfect asymmetry system hence these two cases belong to the different universality regime

the quantum inverse scattering method approach to correlation functions the inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with ther matrix of xxx type these models include the one dimensional bose gas and the heisenberg xxx model the algebraic questions of the problem are considered

dynamical weyl groups and applications abstract following a preceding paper of tarasov and the second author we define and study a new structure which may be regarded as the dynamical analog of the weyl group for lie algebras and of the quantum weyl group for quantized enveloping algebras we give some applications of this new structure

a multidimensional generalization of the barnes integral and the continuous hahn polynomial employing a multidimensional generalization of the beta integral of barnes we derive an expression of the continuous hahn polynomials motivation of the present work is to study such integrals from the viewpoint of a finite difference version of the de rham theory

how to regularize singular vectors and kill the dynamical weyl group abstract let g be a simple lie algebra and let m be the verma module over g with highest weight for a finite dimensional g module u we introduce a notion of a regularizing operator acting in u which makes the meromorphic family of intertwining operators m m u holomorphic and conjugates the dynamical weyl group operators aw end u to constant operators we establish fundamental properties of regularizing operators including uniqueness and prove the existence of a regularizing operator in the case g sl 3

scaling of confined membranes and interfaces we have carried out monte carlo studies of the probability distribution functions pdfs for models of two and three dimensional membranes and interfaces confined between two parallel repulsive walls separated by a distance d for two dimensional interfaces it is known that the position pdf p z for conformally mapped binding potentials scales and is characterized by a universal scaling function p z approximately sin pi z dtheta 1 where theta is the short distance expansion critical exponent for strong weak and intermediate fluctuation regimes our simulation studies show that for a variety of membrane models the pdf has the expected scaling p z u z d d and we find that the same parametrization of the membrane pdfs gives an excellent fit to the numerical data

anomalous decay of pair correlations for two dimensional critical wetting develops a scaling theory for the decay of the spin spin correlation function at two dimensional d 2 wetting transitions in systems with short ranged forces for the critical wetting transition effective hamiltonian results show that the scaling theory is obeyed but the decay of correlations is anomalous in contrast for the complete wetting transition scaling and the ornstein zernike theory are both valid the author argues that the anomalous decay is specific to zero bulk field critical wetting transitions ad d 2

universal and non universal short distance expansions in the strong weak and intermediate fluctuation regimes of critical wetting the author studies the scaling properties of the density profile rho z of a fluid near a wall undergoing a critical wetting transition in the strong fluctuation and in the weak fluctuation scaling regimes a critical exponent theta describing the short distance algebraic decay of the density profile for distances z normal to the wall much less than the perpendicular correlation length zeta perpendicular to is calculated explicitly in dimension d 2 using an effective interfacial hamiltonian model the exponent theta is found to be universal in the strong fluctuation regime theta 1 and in the weak fluctuation regime theta 3 in contrast to the other standard critical exponents theta is non universal at the weak fluctuation mean field borderline the author discusses the origin of the universality of theta in the different fluctuation regimes and derives expressions for theta valid for general d in terms of known critical exponents