makar limanov s conjecture on free subalgebras abstract it is proved that over every countable field k there is a nil algebra r such that the algebra obtained from r by extending the field k contains noncommutative free subalgebras of arbitrarily high rank it is also shown that over every countable field k there is an algebra r without noncommutative free subalgebras of rank two such that the algebra obtained from r by extending the field k contains a noncommutative free subalgebra of rank two this answers a question of makar limanov lenny makar limanov private communication beijing june 2007

difference equations for correlation functions of a 1 n 1 face model with boundary reflection the a 1 n 1 face model with boundary reflection is considered on the basis of the boundary corner transfer matrix bootstrap we construct the fused boundary boltzmann weights to determine the normalization factor we derive difference equations of the quantum knizhnik zamolodchikov type for correlation functions of the boundary model the simplest difference equations are solved in the case of the free boundary condition

free field realization of vertex operators for level two modules of free field realization of vertex operators for level two modules of is shown through the free field realization of the modules given by m idzumi int j mod phys a 9 4449 preprint hep th 9307129 we constructed type i and ii vertex operators when the spin of the associated evaluation module is and type iis for the spin 1

a dimension wise analysis method for the structural acoustic system with interval parameters abstract the interval structural acoustic analysis is mainly accomplished by interval and subinterval perturbation methods potential limitations for these intrusive methods include overestimation or interval translation effect for the former and prohibitive computational cost for the latter in this paper a dimension wise analysis method is thus proposed to overcome these potential limitations in this method a sectional curve of the system response surface along each input dimensionality is firstly extracted the minimal and maximal points of which are identified based on its legendre polynomial approximation and two input vectors i e the minimal and maximal input vectors are dimension wisely assembled by the minimal and maximal points of all sectional curves finally the lower and upper bounds of system response are computed by deterministic finite element analysis at the two input vectors two numerical examples are studied to demonstrate the effectiveness of the proposed method and show that compared to the interval and subinterval perturbation method a better accuracy is achieved without much compromise on efficiency by the proposed method especially for nonlinear problems with large interval parameters

transfer matrix inverison identities for exactly solvable lattice spin models a new functional equation satisfied by the commuting row to row transfer matrices is derived for the eight vertex model functional equations of the same form are satisfied by hard hexagons magnetic hard squares the self dual potts models the andrews baxter forrester models and others these new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices it is conjectured that all solvable models satisfying yang baxter equations possess such inversion identities and that in general these functional equations can be solved for the transfer matrix eigenvalues

a quantum analog of the z algebra we define a natural quantum analog for the z algebra and which we refer to as the zq algebra by modding out the heisenberg algebra from the quantum affine algebra uq sl 2 with level k we discuss the representation theory of this zq algebra in particular we exhibit its reduction to a group algebra and to a tensor product of a group algebra with a quantum clifford algebra when k 1 and k 2 and thus we recover the explicit constructions of uq sl 2 standard modules as achieved by frenkel jing and bernard respectively moreover for arbitrary nonzero level k we show that the explicit basis for the simplest z generalized verma module as constructed by lepowsky and primc is also a basis for its corresponding zq module i e it is invariant under the q deformation for generic q we expect this zq algebra associated with uq sl 2 at level k to play the role of a dynamical symmetry in the off critical zk statistical models

mixing of ground states in vertex models we consider the analogue of the six vertex model constructed from alternating spin n 2 and spin m 2 lines where we identify the transfer matrix and the space on which it acts in terms of the representation theory of we diagonalize the transfer matrix and compute the s matrix we give a trace formula for local correlation functions when n 1 the one point function of a spin m 2 local variable for the alternating lattice with a particular ground state is given as a linear combination of the one point functions of the pure spin m 2 model with different ground states the mixing ratios are calculated exactly and are expressed in terms of irreducible characters of and the deformed virasoro algebra

multi spin correlation functions for the z invariant ising model abstract continuing our work 1 where a explicit formula for the two point functions of the two dimensional z invariant ising model were found i obtain here different results for the higher correlation functions and several consistency checks are done

reformulation of the stochastic bem to improve the computational efficiency in the prediction of the vibro acoustic behaviour of structures with uncertainties abstract non determinism plays a crucial role in the prediction of the vibro acoustic behaviour of structures the stochastic boundary element method sbem allows computing the average response of systems with geometrical uncertainties the method consists in including such perturbations in the standard bem formulation and adding a set of equations representative of the energetic behaviour due to the presence of unknown cross products the addition of a set of auxiliary equation is necessary to solve the problem as a consequence the system dimension becomes very large and the application of the methodology is limited to small models this paper deals with a reformulation of the sbem approach such that the system dimension is drastically decreased and the methodology can be applied to more complex cases moreover the approach is extended to predict the variance of the field variable at no additional computational cost after a theoretical introduction the sbem is applied to two dimensional examples with pressure and velocity boundary conditions the comparison with monte carlo simulations shows a good agreement of the results and a reduction in the computational load is achieved

difference equations for correlation functions of belavin s ℤn symmetric model with boundary reflection belavin s n symmetric elliptic model with boundary reflection is considered on the basis of the boundary ctm bootstrap we find non diagonal k matrices for n 2 that satisfy the reflection equation boundary yang baxter equation and also find non diagonal boltzmann weights for the a 1 n 1 face model even for n 2 we derive difference equations of the quantum knizhnik zamolodchikov type for correlation functions of the boundary model the boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition the resulting quantity is the square of the spontaneous polarization for the bulk n symmetric model up to a phase factor

traces on the sklyanin algebra and correlation functions of the eight vertex model we propose a conjectural formula for correlation functions of the z invariant inhomogeneous eight vertex model we refer to this conjecture as ansatz it states that correlation functions are linear combinations of products of three transcendental functions with theta functions and derivatives as coefficients the transcendental functions are essentially logarithmic derivatives of the partition function per site the coefficients are given in terms of a linear functional tr on the sklyanin algebra which interpolates the usual trace on finite dimensional representations we establish the existence of tr and discuss the connection to the geometry of the classical limit we also conjecture that the ansatz satisfies the reduced qkz equation as a non trivial example of the ansatz we present a new formula for the next nearest neighbour correlation functions

corner transfer matrices in statistical mechanics corner transfer matrices are a useful tool in the statistical mechanics of simple two dimensional models they can be a very effective way of obtaining series expansions of unsolved models and of calculating the order parameters of solved ones here we review these features and discuss the reason why the method fails to give the order parameter of the chiral potts model

functional relations for the order parameters of the chiral potts model following the method of jimbo miwa and others we obtain functional relations for the order parameters of the chiral potts model we have not yet solved these relations here we discuss their properties and show how one should beware of spurious solutions

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

staggered polarization of vertex models with u_q widehat sl n symmetry in this paper we give an explicit formula for level 1 vertex operators related to u q widehat sl n as operators on the fock spaces we derive also their commutation relations as an application we calculate with the vector representation of u q widehat sl n thereby extending the recent work on the staggered polarization of the xxz model

free group algebras in division rings with valuation i abstract let r be an algebra over a commutative ring k suppose that r is endowed with a descending filtration indexed on an ordered group g such that the restriction to k is positive we show that the existence of free algebras on a certain set of generators in the induced graded ring grad r implies the existence of free group algebras in r our best results are obtained for division rings endowed with a valuation

commutation relations of vertex operators for uq sl m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n

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

corner transfer matrix of generalized free fermion vertex systems the hamiltonian limit of the corner transfer matrix ctm of a generalized free fermion vertex system of finite size leads to a quantum spin hamiltonian of the particular form hn sigma n 1n 1 n sigma nx sigma n 1x lambda sigma ny sigma n 1y h sigma nz sigma sub n 1z diagonalization may be achieved for all pairs of parameters lambda h with the use of some new elliptic polynomials which extend the class of special polynomials known so far in the context of ctm

theta function solutions of the quantum knizhnik zamolodchikov bernard equation for a face model we consider the quantum knizhnik zamolodchikov bernard equation for a face model with elliptic weights the sos model we provide explicit solutions as theta functions on the so called combinatorial line in which the model is equivalent to the three colour model these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions

free algebras and free groups in ore extensions and free group algebras in division rings abstract let k be a field of characteristic zero let be an automorphism of k and let be a derivation of k we show that the division ring d k x either has the property that every finitely generated subring satisfies a polynomial identity or d contains a free algebra on two generators over its center in the case when k is finitely generated over a subfield k we then see that for a k algebra automorphism of k and a k linear derivation of k k x having a free subalgebra on two generators is equivalent to having infinite order and k x having a free subalgebra is equivalent to being nonzero as an application we show that if d is a division ring with center k of characteristic zero and d contains a solvable subgroup that is not locally abelian by finite then d contains a free k algebra on two generators moreover if we assume that k is uncountable without any restrictions on the characteristic of k then d contains the k group algebra of the free group of rank two

energy flow variability in a pair of coupled stochastic rods abstract random variations in the transfer functions describing energy flows in a pair of coupled axially vibrating stochastic rods are considered the material properties of the rods are modelled either as random variables or as random processes and the consequent effects on the nature of the power receptance functions are studied by using monte carlo simulation techniques the relevance of these results is discussed in the context of variability in average estimates obtained when using statistical energy analysis sea formalisms the response statistics are shown to vary significantly with the details of the statistical modelling of the system especially in the low and medium frequency ranges moreover the confidence limits show no tendency to converge toward the mean with increasing frequency critical systems the responses of which deviate maximally from the average values are identified and attempts made to understand their nature

the elliptic quantum algebra aq p sln and its bosonization at level one we extend the work of foda et al and propose an elliptic quantum algebra aq p sln similar to the case of aq p sl2 our presentation of the algebra is based on the relation rll llr where r and r are zn symmetric r matrices with the elliptic moduli chosen differently and a scalar factor is also involved with the help of the results obtained by asai et al we realize type i and type ii vertex operators in terms of bosonic free fields for the zn symmetric belavin model we also give a bosonization for the elliptic quantum algebra aq p sln at level one

the critical an 1 1 chain we study the an 1 1 spin chain in the critical regime q 1 we give free boson realizations of the vertex operators and their duals using these free boson realizations we give integral representations for the correlation functions

singularities of n fold integrals of the ising class and the theory of elliptic curves we introduce some multiple integrals that are expected to have the same singularities as the singularities of the n particle contributions n to the susceptibility of the square lattice ising model we find the fuchsian linear differential equation satisfied by these multiple integrals for n 1 2 3 4 and only modulo some primes for n 5 and 6 thus providing a large set of possible new singularities of n we discuss the singularity structure for these multiple integrals by solving the landau conditions we find that the singularities of the associated odes identify up to n 6 with the leading pinch landau singularities the second remarkable obtained feature is that the singularities of the odes associated with the multiple integrals reduce to the singularities of the odes associated with a finite number of one dimensional integrals among the singularities found we underline the fact that the quadratic polynomial condition 1 3w 4w2 0 that occurs in the linear differential equation of 3 actually corresponds to a remarkable property of selected elliptic curves namely the occurrence of complex multiplication the interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group namely isogenies of elliptic curves is sketched most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations suggesting an interpretation in terms of motivic mathematical structures beyond the theory of elliptic curves

exactly solved models on planar graphs with vertices in z 3 it is shown how exactly solved edge interaction models on the square lattice may be extended onto more general planar graphs with edges connecting a subset of next nearest neighbour vertices of mathbb z 3 this is done by using local deformations of the square lattice that arise through the use of the star triangle relation similar to baxter s z invariance property these local deformations leave the partition function invariant up to some simple factors coming from the star triangle relation the deformations used here extend the usual formulation of z invariance by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model the quasi classical limit is also considered in which case the deformations imply a classical z invariance property as well as a related local closure relation for the action functional of a system of classical discrete laplace equations

bosonization of vertex operators for the face model we present a free boson realization of the vertex operators and their duals for the solvable sos lattice model of type we discuss a possible connection with the calculation of correlation functions

difference equations for the correlation functions of the eight vertex model the authors propose that the correlation functions of the inhomogeneous eight vertex model in the anti ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters solving the simplest difference equation they obtain the expression for the spontaneous staggered polarization conjectured by baxter and kelland 1974 they also discuss a related construction of vertex operators on the lattice

corner transfer matrices abstract for two dimensional lattice models with interactions only between nearest and diagonally nearest neighbour spins a well known concept is the row to row transfer matrix less well known is the corner transfer matrix ctm this has some very useful properties if it is normalized so that its largest eigenvalue is unity and the eigenvalues are arranged in numerically decreasing order then each eigenvalue tends to a limit as the lattice becomes large for those models which have been solved exactly notably the ising eight vertex and hard hexagon models this limiting eigenvalue distribution is very simple being basically that of a direct product of two by two matrices from it the order parameter can easily be obtained for all models one can write down formally exact matrix relations for the ctm but the matrices are of infinite size if one uses a representation in which the ctm is diagonal and then truncates these relations to finite size then one obtains a quite accurate approximation the larger the size the greater the accuracy i g enting and i have thereby obtained comparatively long series expansions for the ising model in a field and for the hard squares model

corner transfer matrices and corner magnetization for the ising model the authors consider the corner magnetization of an ising square lattice with free edges along a diagonal direction using some remarkable properties of the corner transfer matrix they give a unified treatment for various opening angles the calculation of the magnetization is reduced to the solution of a matrix equation in the thermodynamic limit they construct an explicit form of this equation and discuss its numerical and asymptotic solution and the results obtained thereby

spontaneous polarization of the kondo problem associated with the higher spin six vertex model we study the multi channel kondo model associated with an integrable higher spin analogue of the anti ferroelectric six vertex model which is constructed by inserting spin to spin 1 lines we formulate the problem in terms of representation theory of quantum affine algebra 1 we derive an exact formula for the spontaneous staggered polarization for our model which corresponds to baxter s formula 2 for the six vertex model

a vertex operator approach for correlation functions of belavin s ℤ nℤ symmetric model belavin s n symmetric model is considered on the basis of bosonization of vertex operators in the a 1 n 1 model and vertex face transformation the corner transfer matrix ctm hamiltonian of the n symmetric model and tail operators are expressed in terms of bosonized vertex operators in the a 1 n 1 model correlation functions of the n symmetric model can be obtained by using these objects in principle in particular we calculate spontaneous polarization which reproduces the result we obtained in 1993

functional relations for the order parameters of the chiral potts model low temperature expansions abstract this is the third in a series of papers in which we set up and discuss the functional relations for the split rapidity line correlation function in the n state chiral potts model the order parameters of the model can be obtained from this function here we consider the case n 3 and write the equations explicitly in terms of the hyperelliptic functions parametrization we also present four term low temperature series expansions which we hope will cast light on the analyticity properties needed to solve the relations the problem remains unsolved but we hope that this will prove to be a step in the right direction

an integral formulation with random parameters adapted to the study of the vibrational behaviour of structures in the middle and high frequency field the fundamentals of a new boundary element formulation applied to the study of structural dynamics on a wide frequency range is proposed in this paper random geometrical parameters are introduced to the integral representations in order to model the increasing sensitivity of the harmonic vibrational responses to any parameter perturbation when the frequency increases it is shown that the significant unknowns of the formulation are the expectations of the square boundary kinematic unknowns indeed these variables are able to describe accurately the responses of the structures in the low frequency field and deliver a smooth response in the middle and high frequency field corresponding to the global trend of the structural dynamic response in this frequency domain the formulation is established for isolated and assembled one two and three dimensional structures numerical applications illustrate the effectiveness of the method

the order parameter of the chiral potts model an outstanding problem in statistical mechanics is the order parameter of the chiral potts model an elegant conjecture for this was made in 1983 it has since been successfully tested against series expansions but there is as yet no proof of the conjecture here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy then one can indeed verify the conjecture the method is based on the broken rapidity line approach pioneered by jimbo et al j phys a 26 2199 2210 1993

free subalgebras of division algebras over uncountable fields we study the existence of free subalgebras in division algebras and prove the following general result if a is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0 then either the quotient division algebra of a contains a free algebra on two generators or it is left algebraic over every maximal subfield as an application we prove that if k is an uncountable algebraically closed field and a is a finitely generated k algebra that is a domain of gk dimension strictly less than 3 then either a satisfies a polynomial identity or the quotient division algebra of a contains a free k algebra on two generators

zamolodchikov faddeev algebra related to zn symmetric elliptic r matrix we give representations for the zamolodchikov faddeev algebra zfa in the elliptic case and prove the self consistency of this algebra in such a case we also study the implications of our approach for those algebras which are relevant to the extension of the q deformed affine an 1 algebra

response variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea of high frequency noise and vibration a complex engineering structure is represented as an assembly of subsystems the response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem and these energies are found by employing the principle of power balance strictly the computed energy is an average taken over an ensemble of random structures and for many years there has been interest in extending the sea prediction to the variance of the energy a variance prediction method for a general built up structure is presented here closed form expressions for the variance are obtained in terms of the standard sea parameters and an additional set of parameters alpha k that describe the nature of the power input to each subsystem k and alpha ks that describe the nature of the coupling between subsystems k and s the theory is validated by comparison with monte carlo simulations of plate networks and structural acoustic systems

determinant representation for correlation functions of spin 1 2 xxx and xxz heisenberg magnets we consider zero temperature correlation functions of the spin 1 2 xxz heisenberg chain in the critical regime 1 1 in a magnetic field starting from the algebraic bethe ansatz we derive representations for various correlation functions in term of determinats of fredholm integral operators

a new random boundary element formulation applied to high frequency phenomena the basis of a new boundary element method adapted to wide frequency range applications is proposed in this paper the formulation employs the frequency domain dynamic fundamental solutions of the problems random geometrical parameters are introduced in the integral equations these equations are then modified to exhibit the products of the different kinematic variables the expectations of the new equations with respect to the latter random variables are considered and the new variables of the formulation are the different stochastic moments force displacement variables this formulation is applied to different structures such as beams rods and assembled one dimensional systems

a system of difference equations with elliptic coefficients and bethe vectors an elliptic analogue of theq deformed knizhnik zamolodchikov equations is introduced a solution is given in the form of a jackson type integral of bethe vectors of the xyz type spin chains

corner transfer matrices of the chiral potts model ii the triangular lattice we consider a two dimensional edge interaction model satisfying the star triangle relations for the triangular lattice the corner transfer matrices are functions of three rapidities we show that they possess various factorization properties and satisfy certain equations we indicate how these equations can be solved for the ising model we then consider the three state chiral potts model and obtain low temperature solutions to the equations the conjectured formula for the order parameter the spontaneous magnetization is verified to one more order in a series expansion

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

a quantum algebra approach to basic multivariable special functions generalizations to a base q of the appell and lauricella hypergeometric functions are studied within the framework of the representation theory of quantum algebras of type an this allows deriving various identities and formulae involving these q functions

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

free groups in normal subgroups of the multiplicative group of a division ring abstract let d be a division ring with center z and multiplicative group d 0 d and let n be a normal subgroup of d we investigate various conditions under which n must contain a free noncyclic subgroup in one instance assuming that the transcendence degree of z over its prime field is infinite and that n contains a nonabelian solvable subgroup we use a construction method due to chiba to exhibit free generators of the free subgroup

corner transfer matrices and lorentz invariance on a lattice abstract the continuum limit of the baxter eight vertex lattice model is the lorentz invariant massive thirring sine gordon field theory here it is shown that the baxter model exhibits a continuous symmetry which is an exact lattice generalization generalization of lorentz invariance and that the corner transfer matrix recently developed by baxter is a lattice boost operator the role of elliptic function parameters as lattice rapidity variables is discussed

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

conformal invariance and critical spectrum of corner transfer matrices the finite size scaling spectra of ising model corner transfer matrices and their generators are studied at criticality the generators are diagonalised using fermion algebra the low lying eigenvalues given by the zeros of meixner polynomials are equally spaced and collapse like 1 log n for large n as predicted by conformal invariance the asymptotics are obtained using a generalised euler maclaurin summation formula the shift in the largest eigenvalue is given analytically as pi c 6 log n with central charge c 1 2 the spectrum generating functions for both fixed and free boundary conditions are expressed simply in terms of the c 1 2 virasoro characters chi delta q with modular parameter q exp pi log n and conformal dimensions delta 0 1 2 1 16

extended z invariance for integrable vector and face models and multi component integrable quad equations in a previous paper kels in j phys a 50 49 495202 2017 the author has established an extension of the z invariance property for integrable edge interaction models of statistical mechanics that satisfy the star triangle relation str form of the yang baxter equation ybe in the present paper an analogous extended z invariance property is shown to also hold for integrable vector models and interaction round a face irf models of statistical mechanics respectively as for the previous case of the str the z invariance property is shown through the use of local cubic type deformations of a 2 dimensional surface associated to the models which allow an extension of the models onto a subset of next nearest neighbour vertices of mathbb z 3 while leaving the partition functions invariant these deformations are permitted as a consequence of the respective ybe s satisfied by the models the quasi classical limit is also considered and it is shown that an analogous z invariance property holds for the variational formulation of classical discrete laplace equations which arise in this limit from this limit new integrable 3d consistent multi component quad equations are proposed which are constructed from a degeneration of the equations of motion for irf boltzmann weights

bethe subalgebras in affine birman murakami wenzl algebras and flat connections for q kz equations commutative sets of jucys murphy elements for affine braid groups of types were defined construction of r matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the type jucys murphy elements are given we describe a general method to produce flat connections for the two boundary quantum knizhnik zamolodchikov equations as necessary conditions for sklyanin s type transfer matrix associated with the two boundary multicomponent zamolodchikov algebra to be invariant under the action of the type jucys murphy elements we specify our general construction to the case of the birman murakami wenzl algebras bmw algebras for short as an application we suggest a baxterization of the dunkl cherednik elements in the double affine hecke algebra of type a

the riemann surface of the chiral potts model free energy function in a recent paper we derived the free energy or partition function of the n state chiral potts model by using the infinite lattice inversion relation method together with a non obvious extra symmetry this gave us three recursion relations for the partition function per site t pq of the infinite lattice here we use these recursion relations to obtain the full riemann surface of t pq in terms of the t p t q variables it consists of an infinite number of riemann sheets each sheet corresponding to a point on a 2n 1 dimensional lattice for n 2 the function t pq is meromorphic on this surface we obtain the orders of all the zeros and poles for n odd we show that these orders are determined by the usual inversion and rotation relations without the extra symmetry together with a simple linearity ansatz for n even this method does not give the orders uniquely but leaves only n 4 4 parameters to be determined

the elliptic gamma function and sl 3 z z3 abstract the elliptic gamma function is a generalization of the euler gamma function and is associated to an elliptic curve its trigonometric and rational degenerations are the jackson q gamma function and the euler gamma function respectively the elliptic gamma function appears in baxter s formula for the free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qkzb equations in this paper the properties of this function are studied in particular we show that elliptic gamma functions are generalizations of automorphic forms of g sl 3 z z 3 associated to a non trivial class in h3 g z

corner transfer matrices for the ising model we consider corner transfer matrices for the regular ising model and show that finding a diagonal representation is a soluble problem in interacting fermions we show how the requirement of having an invertible diagonalising transformation determines which representation must be used and explain why baxter s original solution of the problem did not produce such an invertible transformation

vertex operators in solvable lattice models the basic properties of q vertex operators are formulated in the context of the andrews baxter forrester abf series as an example of face interaction models the q difference equations satisfied by their correlation functions are derived and their connection with representation theory established the q difference equations of the kashiwara miwa km series are discussed as an example of edge interaction models next the ising model the simplest special case of both abf and km series is studied in more detail using the jordan wigner fermions in particular all matrix elements of vertex operators are calculated

corner transfer matrix of a critical free fermion system in order to diagonalize the generator of the corner transfer matrix of a critical free fermion 8 vertex system with magnetic field the authors derive an integral representation for the eigenvector components of the generator at large n they numerically verify the predictions of conformal invariance for critical systems

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