spectral parameter dependent approach to quantized algebra its multiparameter deformations and their q oscillator realizations yang baxterization of faddeev reshetikhin takhtajan algebra 1989 leading to quantum ybe is considered for a class of r matrices which satisfy hecke relations the authors apply such construction of the gl n case together with its multiparameter deformations revealing the connection between frt relations and the extended trigonometric sklyanin algebra new realizations of frt algebra through q oscillator modes are presented and their potential importance in the context of quantum integrable models is discussed a multiparametrized r matrix with the inclusion of spectral parameter is constructed as a by product

three dimensional integrable models and associated tangle invariants in this paper we show that the boltzmann weights of the three dimensional baxter bazhanov model give representations of the braid group if some suitable spectral limits are taken in the trigonometric case we classify all possible spectral limits which produce braid group representations furthermore we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones

the limits of the solutions of a nonautonomous linear delay difference equation abstract consider the system of linear delay difference equations where the coefficients a j n are square matrices and k j and l j are nonnegative integers in this note we show that if the coefficients are small then every solution of the above equation tends to a constant vector as n and the value of the limit can be characterized by a special solution of the matrix equation and the initial conditions

a new quasitriangular hopf algebra as the nontrivial quantum double of a simplest c algebra and its universal r matrix for yang baxter equation endowing a simple c algebra generated by two commuting elements and a unit with a noncocommutative hopf algebra structure a new quantum double and the corresponding universal r matrix for the yang baxter equation is constructed the finite dimensional representations of the quantum double is also studied and a concrete r matrix is presented as an example

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

solutions of yang baxter equations with colours standard and nonstandard colour extended solutions of ybe associated with g bn cn and dn are derived and the coloured birman wenzl algebra cb wa is established the authors point out that the cb wa still satisfies the redundance conditions of murakami 1989 for nonstandard solution they take the nontrivial colour solution of d2 as an example they find that the colour constraints induce a new solution with q4 1 the implication of this is not clear

cyclic monodromy matrices for sl n trigonometric r matrices the algebra of monodromy matrices forsl n trigonometricr matrix is studied it is shown that a generic finite dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product ofl operators cocommutativity of representations is discussed and intertwiners for factorizable representations are written through the boltzmann weights of thesl n chiral potts model

an su 2 analogue of the azbel hofstadter hamiltonian motivated by the recent findings of wiegmann zabrodin and faddeev kashaev concerning the appearance of the quantum symmetry in the problem of a bloch electron on a two dimensional magnetic lattice we introduce a modification of the tight binding azbel hofstadter hamiltonian that is a specific spin s euler top that can be considered as its classical analogue the eigenvalue problem for the proposed model in the coherent state representation is described by the s gap lame equation and thus is completely solvable we observe a striking similarity between the shapes of the spectra of the two models for various values of the spin s

solutions of the yang baxter equation with extra nonadditive parameters ii uq gl m n the type i quantum superalgebras are known to admit non trivial one parameter families of inequivalent finite dimensional irreps even for generic q we apply the recently developed technique to construct new solutions to the quantum yang baxter equation associated with the one parameter family of irreps of uq gl m n thus obtaining r matrices that depend not only on a spectral parameter but in addition on further continuous parameters these extra parameters enter the yang baxter equation in a similar way to the spectral parameter but in a non additive form

the drinfeld double gl n tn we present a self dual drinfeld double structure underlying the an series of simple lie algebras such double is constructed through a central extension tn of gl n and is obtained by pairing two disjoint solvable subalgebras coming from positive and negative roots the cartan weyl basis of gl n is shown to be completely determined by the compatibility conditions in the double a natural lie bialgebra structure on gl n is obtained that offers a new perspective for the construction of its quantum deformations

the universal r matrix and the yang baxter equation with parameters the authors formulate and demonstrate with some examples a method for obtaining solutions of the yang baxter equation with parameters

exotic quantum double its universal r matrix and their representations the exotic quantum double and its universal r matrix for the quantum yang baxter equation are constructed in terms of drinfeld s quantum double theory as a new quasitriangular hopf algebra it is different from those standard quantum doubles that are the q deformations for lie algebras or lie superalgebras by studying its representation theory many parameter representations of the exotic quantum double are obtained with an explicit example associated with lie algebra a2 the multiparameter r matrices for the quantum yang baxter equation can result from the universal r matrix of this exotic quantum double and these representations

a few endpoint geodesic restriction estimates for eigenfunctions we prove a couple of new endpoint geodesic restriction estimates for eigenfunctions in the case of general 3 dimensional compact manifolds after a tt argument simply by using the l 2 boundedness of the hilbert transform on mathbb r we are able to improve the corresponding l 2 restriction bounds of burq gerard and tzvetkov duke math j 138 445 486 2007 and hu forum math 6 1021 1052 2009 also in the case of 2 dimensional compact manifolds with nonpositive curvature we obtain improved l 4 estimates for restrictions to geodesics which by holder s inequality and interpolation implies improved l p bounds for all exponents p 2 we do this by using oscillatory integral theorems of hormander ark mat 11 1 11 1973 greenleaf and seeger j reine angew math 455 35 56 1994 and phong and stein int math res notices 4 49 60 1991 along with a simple geometric lemma lemma 3 2 about properties of the mixed hessian of the riemannian distance function restricted to pairs of geodesics in riemannian surfaces we are also able to get further improvements beyond our new results in three dimensions under the assumption of constant nonpositive curvature by exploiting the fact that in this case there are many totally geodesic submanifolds

the behavior of solutions of linear volterra difference equations with infinite delay a class of linear volterra difference equations with infinite delay is considered a basic theorem on the behavior of solutions is established and as a corollary a useful exponential estimate for solutions is obtained and a stability criterion is derived these results are achieved via an appropriate positive root of the corresponding characteristic equation

periodic first order linear neutral delay differential equations some new asymptotic and stability results are given for a first order linear neutral delay differential equation with periodic coefficients and constant delays the asymptotic behavior of the solutions and the stability of the trivial solution are described by the use of an appropriate real root of an equation which is in a sense the corresponding characteristic equation

new constant and trigonometric 4 4 solutions to the yang baxter equations a classification of triangular constant solutions of the yang baxter equations is performed the formulae for the baxterization of braid group representation are used for finding trigonometric solutions of the yang baxter equations from the constant ones some of the obtained solutions correspond to known solutions but several new ones are obtained as well the calculations confirm that the baxterization formulae for more than two eigenvalues of the r matrix are not universal

on the behavior of the solutions to periodic linear delay differential and difference equations some new results on the behavior of the solutions to periodic linear delay differential equations as well as to periodic linear delay difference equations are given these results are obtained by the use of two distinct roots of the corresponding so called characteristic equation

the behavior of solutions of linear integro differential equations with unbounded delay abstract a basic theorem on the behavior of solutions of scalar linear integro differential equations with unbounded delay is established as a consequence of this theorem a stability criterion is obtained

universal spectral parameter dependent lax operators for the drinfeld double of the dihedral group d3 two universal spectral parameter dependent lax operators are presented in terms of the elements of the drinfeld double d d3 of the dihedral group d3 applying representations of d d3 to these yields matrix solutions of the yang baxter equation with a spectral parameter

zn n 1 generalization of the chiral potts model we show that ther matrix which intertwines twon by nn 1 state cyclicl operators related with a generalization ofuq sl n algebra can be considered as a boltzmann weight of four spin box for a lattice model with two spin interaction just as ther matrix of the checkerboard chiral potts model the rapidity variables lie on the algebraic curve of the genusg n2 n 1 n 1 n n 1 defined by 2n 3 independent moduli this curve is a natural generalization of the curve which appeared in the chiral potts model factorization properties of thel operator and its connection to the sos models are also discussed

the vertex formulation of the bazhanov baxter model in this paper we formulate an integrable model on the simple cubic lattice then valued spin variables of the model belong to edges of the lattice the boltzmann weights of the model obey the vertex type tetrahedron equation in the thermodynamic limit our model is equivalent to the bazhanov baxter model in the case whenn 2 we reproduce korepanov s and hietarinta s solutions of the tetrahedron equation as special cases

trigonometric yang baxterization of coloured r matrix for a given coloured solution r lambda mu of the quantum yang baxter equation in appropriate form the authors present an explicit prescription to generate trigonometric solutions r lambda mu x of the yang baxter equation with spectral parameters x when the given r lambda mu admits the birman wenzl algebraic structure with colour they find that two types of the r lambda mu x are generated by such r lambda mu new explicit examples associated with the fundamental representations of an bn cn and dn are given in terms of this prescription

generalized chiral potts models and minimal cyclic representations of u_q widehat mathfrak g mathfrak l n c finite dimensional cyclic representations of the quantized universal enveloping algebra u q g exist only when the deformation parameter q is a root of unity q n 1 in such representations the n th power of each chevalley generator acts as a non zero scalar these values of e i n f i n and t i n are part of a continuous family of parameters upon which the cyclic representations depend

extended birman wenzl algebra and yang baxterization a new algebra a is presented by directly extending the birman wenzl algebra if a braid group representation with four distinct eigenvalues satisfies a then it can be yang baxterized to derive the solution of the yang baxter equation in terms of a standard scheme

construction of r matrices for symmetric tensor representations related to in this paper we construct a new factorized representation of the r matrix related to the affine algebra for symmetric tensor representations with arbitrary weights using the 3d approach we obtain explicit formulas for the matrix elements of the r matrix and give a simple proof that a twisted r matrix is stochastic we also discuss symmetries of the r matrix its degenerations and compare our formulas with other results available in the literature

l p norms nodal sets and quantum ergodicity abstract for small range of p 2 we improve the l p bounds of eigenfunctions of the laplacian on negatively curved manifolds our improvement is by a power of logarithm for a full density sequence of eigenfunctions we also derive improvements on the size of the nodal sets our proof is based on a quantum ergodicity property of independent interest which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

star triangle relation for a three dimensional model the solvablesl n chiral potts model can be interpreted as a three dimensional lattice model with local interactions to within a minor modification of the boundary conditions it is an ising type model on the body centered cubic lattice with two and three spin interactions the corresponding local boltzmann weights obey a number of simple relations including a restricted star triangle relation which is a modified version of the well known star triangle relation appearing in two dimensional models we show that these relations lead to remarkable symmetry properties of the boltzmann weight function of an elementary cube of the lattice related to the spatial symmetry group of the cubic lattice these symmetry properties allow one to prove the commutativity of the row to row transfer matrices bypassing the tetrahedron relation the partition function per site for the infinite lattice is calculated exactly

a new q deformed boson realization of quantum algebra slq n 1 and nongeneric slq 2 r matrices in this paper a new q deformed boson realization of quantum algebra slq n 1 is presented for the first time and its representations are obtained in the nongeneric case that q is a root of unity a new type of slq 2 r matrices are systematically constructed through the universal r matrix

a note on graded yang baxter solutions as braid monoid invariants we construct two osp n mod 2m solutions of the graded yang baxter equation by using the algebraic braid monoid approach the factorizable s matrix interpretation of these solutions is also discussed

l operator for belavin s r matrix acting on the space of theta functions for the z nz symmetric elliptic r matrix l operators which act on the space of a 1 n 1 integrable characters or the affine weyl group invariants of the fixed level l are constructed for l 1 the matrix elements of the l operator and those of the r matrix are shown to be the same

crossing symmetry in elliptic solutions of the yang baxter equation and a new l operator for belavin s solution some algebraic structures in elliptic solutions of the yang baxter equations are investigated the author proves the crossing symmetry in belavin s model 1981 as well as in the an 1 1 face model and constructs a new family of l operators for belavin s r matrix as an application

improved critical eigenfunction restriction estimates on riemannian manifolds with constant negative curvature abstract we show that one can obtain logarithmic improvements of l 2 geodesic restriction estimates for eigenfunctions on 3 dimensional compact riemannian manifolds with constant negative curvature we obtain a log 1 2 gain for the l 2 restriction bounds which improves the corresponding bounds of burq gerard and tzvetkov 4 hu 10 chen and sogge 6 we achieve this by adapting the approaches developed by chen and sogge 6 blair and sogge 3 xi and the author 19 we derive an explicit formula for the wave kernel on 3d hyperbolic space which improves the kernel estimates from the hadamard parametrix in chen and sogge 6 we prove detailed oscillatory integral estimates with fold singularities by phong and stein 12 and use the poincare half space model to establish bounds for various derivatives of the distance function restricted to geodesic segments on the universal cover h 3

nonlinear waves on 3d hyperbolic space in this article global in time dispersive estimates and strichartz estimates are explored for the wave equation on three dimensional hyperbolic space due to the negative curvature extra dispersion is noted as compared to the euclidean case and a wider range of strichartz estimates is proved using these small data global existence to semilinear wave equations is shown for a range of powers that is broader than that known for euclidean space

the dynamical yang baxter relation and the minimal representation of the elliptic quantum group in this article we give the general forms of the minimal l matrix the elements of the l matrix are c numbers associated with the boltzmann weights of the an 11 interaction round a face irf model and the minimal representation of the an 1 series elliptic quantum group given by felder and varchenko the explicit dependence of elements of l matrices on spectral parameter z are given they are of five different forms a 1 4 and b the algebra for the coefficients which do not depend on z are given the algebra of form a is proved to be trivial while that of form b obeys yang baxter equation ybe we also give the pbw base and the centers for the algebra of form b

exactly solvable quantum spin ladders associated with the orthogonal and symplectic lie algebras we extend the results of spin ladder models associated with the lie algebras su 2 n to the case of the orthogonal and symplectic algebras o 2 n sp 2 n where n is the number of legs for the system two classes of models are found whose symmetry either orthogonal or symplectic has an explicit n dependence integrability of these models is shown for an arbitrary coupling of xx type rung interactions and applied magnetic field term

ruijsenaars macdonald type difference operators from belavin model with open boundary conditions antisymmetric fusions of the belavin model with integrable open boundary conditions are studied the centre of the corresponding operator algebra the sklyanin determinant is constructed we prove that the transfer matrices of the fusion hierarchies mutually commute utilizing the factorized l operators this commuting family is expressed as mutually commuting difference operators acting on a function space this gives ruijsenaars macdonald type commuting difference operators which describe the conservation laws of an integrable system

affine toda field theory as a 3 dimensional integrable system the affine toda field theory is studied as a 2 1 dimensional system the third dimension appears as the discrete space dimension corresponding to the simple roots in the an affine root system enumerated according to the cyclic order on the an affine dynkin diagram we show that there exists a natural discretization of the affine toda theory the quantum analog of the variables is found the thermodynamic bethe ansatz of the affine toda system is studied in the limit l n it is shown that the free energy of the systems grows proportionally to the volume

new solutions of yang baxter equations birman wenzl algebra and quantum group structures new braid group representations associated with the fundamental representations of bn cn and dn are derived they are shown to satisfy birman wenzl algebra by yang baxterization the authors obtain new solutions of yang baxter equations including the twisted extensions quantum group structures related to these new solutions are explicitly given

thermodynamic limit for a spin lattice an integrable spin lattice is a higher dimensional generalization of integrable spin chains in this paper we consider a special spin lattice related to quantum mechanical interpretation of the three dimensional lattice model in statistical mechanics zamolodchikov and baxter the integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables the significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters there is a specific bilinear functional equation for the eigenvalues of this transfer matrix the spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity the limiting form of bilinear equation is derived it allows to analyze the distributions of eigenvalues of the whole commuting set the ground state distribution is obtained explicitly a structure of excited states is discussed

the cyclic boson operators and new representations of the quantum algebra slq 2 for q a root of unity the concept of the cyclic boson operator is introduced for the explicit construction of new representations of the quantum algebra slq 2 when q is a root of unity the method used can easily be generalized to other quantum algebras or quantum superalgebras

on quantization of z2 graded algebras quantization of z2 graded lie bialgebras viewed as a generalization of sl 2 is investigated a method of building multiparameter solutions is developed we study the role of the quantum double construction in the selected class and prove that a quantized coboundary z2 graded lie bialgebra contains a quasitriangular hopf subalgebra

a non standard quantum double as the l state boson algebra and its r matrices for the quantum yang baxter equation in this paper we construct a new quantum double by endowing the l state boson algebra with a non trivial hopf algebra structure which is not a q deformation of the lie algebra or superalgebra the universal r matrix for the yang baxter equation associated with this new quantum group structure is obtained explicitly by building the representations of this quantum double we obtain some r matrices which can result in new representations of the braid group

the r operator for a modular double we construct the r operator the solution of the yang baxter equation acting in the tensor product of two infinite dimensional representations of faddeev s modular double this r operator intertwines the product of two l operators associated with the modular double and it is built from three basic operators generating the permutation group of four parameters

ansatz of hans bethe for a two dimensional lattice bose gas the method of q oscillator lattices proposed recently in bazhanov and sergeev 2005 preprint hep th 0509181 provides the tool for a construction of various integrable models of quantum mechanics in 2 1 dimensional spacetime in contrast to any one dimensional quantum chain its twodimensional generalizations quantum lattices admit different geometrical structures in this paper we consider the q oscillator model on a special lattice the model may be interpreted as a two dimensional lattice bose gas the most remarkable feature of the model is that it allows the coordinate bethe ansatz the p particles wavefunction is the sum of plane waves consistency conditions is the set of 2p equations for p one particle wave vectors these bethe ansatz equations are the main result of this paper

non birman wenzl algebraic properties and redundancy of exotic enhanced yang baxter operator for spin model in this paper we explicitly construct the markov trace for the general coloured exotic braid group representations bgr with spin j it is verified that the bgr for j 1 is redundant in the sense of murakami 1990 but no birman wenzl algebra

generalized bethe ansatz equations for hofstadter problem the problem of diagonalization of the quantum mechanical hamiltonian governing dynamics of an electron on a two dimensional triangular or square lattice in external uniform magnetic field applied perpendicularly to the lattice plane the flux through lattice cell divided by the elementary quantum flux being a rational number is reduced to the generalized bethe ansatz like equations on the high genus algebraic curve our formulae for the trigonometric case where the genus of the curve vanishes contain as a particular case a recent result of wiegmann and zabrodin

knot invariants associated with a particular n continuous limit of the baxter bazhanov model first we briefly recall the definition of the three dimensional baxter bazhanov lattice model the spins of this model are elements ofzn and ther matrix is associated to the algebrauqsl n ifq is a primitiventh root of unity then we construct a particularn limit of the model in which it is meaningful to interpret the spins as elements ofr and which gives the free gaussian boson model finally we study special limits of the rapidity variables in which we obtain braid group representations and we show that forn odd the associated knot invariants are given by the inverse of products of alexander polynomials evaluated at certain roots of unity

truncated global transfer matrix trigonometric goryachev chaplygin gyrostat the rtt relation with truncation is solved for the r matrix associated with the six vertex model it turns out that the finite expansion of the global solution of the transfer matrix t u in terms of the spectral parameter u can be realized by the defined algebra formed by and the non commutative coordinates the hamiltonian of the system has been found which is the trigonometric deformation of the goryachev chaplygin gyrostat

r matrices of three state hamiltonians solvable by coordinate bethe ansatz we review some of the strategies that can be implemented to infer an r matrix from the knowledge of its hamiltonian we apply them to the classification achieved in crampe frappat and ragoucy j phys a 46 405001 2013 on three state u 1 invariant hamiltonians solvable by coordinate bethe ansatz focusing on models for which the s matrix is not trivial for the 19 vertex solutions we recover the r matrices of the well known zamolodchikov fateev and izergin korepin models we point out that the generalized bariev hamiltonian is related to both main and special branches studied by martins in nucl phys b 874 243 2013 that we prove to generate the same hamiltonian the 19 vertex spr model still resists to the analysis although we are able to state some no go theorems on its r matrix for 17 vertex hamiltonians we produce a new r matrix

asymptotic behavior of a nonhomogeneous linear recurrence system consider the nonhomogeneous linear recurrence systemxn 1 a bn xn gn where a and bn n 0 1 are square matrices and gn n 0 1 are column vectors in this paper we describe in terms of the initial condition the asymptotic behavior of the solutions of this equation in the case when a has a simple dominant eigenvalue 0 n 0 bn and n 0 0 n gn the proof is based on the duality between the solutions of the above equation and the solutions of the associated adjoint equation as a consequence we obtain a similar result for higher order scalar equations

cyclic l operator related with a 3 state r matrix we consider the problem of constructing a cyclicl operator associated with a 3 stater matrix related to theuq sl 3 algebra atqn 1 this problem is reduced to the construction of a cyclic i e with no highest weight vector representation of some twelve generating element algebra which generalizes theuq sl 3 algebra we found such representation acting incn cn cn the necessary conditions of the existence of the intertwining operator for two representations are also discussed

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

improved critical eigenfunction restriction estimates on riemannian surfaces with nonpositive curvature we show that one can obtain improved l4 geodesic restriction estimates for eigenfunctions on compact riemannian surfaces with nonpositive curvature we achieve this by adapting sogge s strategy in improved critical eigenfunction estimates on manifolds of nonpositive curvature preprint we first combine the improved l2 restriction estimate of blair and sogge concerning toponogov s theorem and logarithmic improvement of estimates of eigenfunctions preprint and the classical improved l infty estimate of berard to obtain an improved weak type l4 restriction estimate we then upgrade this weak estimate to a strong one by using the improved lorentz space estimate of bak and seeger math res lett 18 4 767 781 2011 this estimate improves the l4 restriction estimate of burq et al duke math j 138 445 486 2007 and hu forum math 6 1021 1052 2009 by a power of log log lambda 1 moreover in the case of compact hyperbolic surfaces we obtain further improvements in terms of log lambda 1 by applying the ideas from chen and sogge commun math phys 329 3 435 459 2014 and blair and sogge concerning toponogov s theorem and logarithmic improvement of estimates of eigenfunctions preprint we are able to compute various constants that appeared in chen and sogge commun math phys 329 3 435 459 2014 explicitly by proving detailed oscillatory integral estimates and lifting calculations to the universal cover mathbb h 2

finite dimensional irreducible representations of the quantum supergroup uq gl m n finite dimensional irreducible representations of the quantum supergroup uq gl m n at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction the representation theory is rather similar to that of gl m n at generic q but drastically different when q is a root of unity in the latter case atypicality conditions of highest weight irreducible representations irreps are substantially altered and such finite dimensional irreps arise that do not have highest weight and or lowest weight vectors as concrete examples the irreps of uq gl 2 1 are classified

integrable boundary conditions for multi species asep the first result of the present paper is to provide classes of explicitsolutions for integrable boundary matrices for the multi species asep with anarbitrary number of species all the solutions we have obtained can be seen as representations of a newalgebra that contains the boundary hecke algebra the boundary hecke algebra isnot sufficient to build these solutions this is the second result of ourpaper

yang baxterization of braid group representations for a given braid group representation bgr a process of the yang baxterization is formulated to generate solutions of the yang baxter equation ybe when a bgr admits the birman wenzl bw algebraic structure this process can be explicitly passed through and two types of trigonometric solutions of ybe are generated from such a bgr these two solutions have the essential difference to each other and both of them preserve the crossing symmetry property if the given bgr has by taking certain reduction on the bw algebra the rational solution is also generated a practical condition to judge whether a bgr satisfies the bw algebra is given from which one finds that not only the familiar bgrs of 5 7 9 but also some new ones obtained recently in 12 have the bw structure thus they can be explicitly yang baxterized to solutions of the ybe

tetrahedron equations boundary states and the hidden structure of mathscr u _q left d_n 1 right simple periodic 3d 2d compactification of the tetrahedron equations gives the yang baxter equations for various evaluation representations of in this paper we construct an example of fixed non periodic 3d boundary conditions producing a set of yang baxter equations for these boundary conditions resemble a fusion in the hidden direction

tetrahedron equation and generalized quantum groups we construct families of solutions of the yang baxter equation from n products of three dimensional r and l operators satisfying the tetrahedron equation they are identified with the quantum r matrices for the hopf algebras known as generalized quantum groups depending on the number of r s and l s involved in the product the trace construction interpolates the symmetric tensor representations of and the antisymmetric tensor representations of whereas a boundary vector construction interpolates the q oscillator representation of and the spin representation of the intermediate cases are associated with an affinization of quantum superalgebras

exactly soluble isotropic spin ladder models the undeformed limit of the dilute two colour braid monoid algebra gives a natural basis for the description of spin ladder models and allows different baxterization ansatze based on this observation we find the complete classification of exactly soluble generalized isotropic spin ladder models

representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the yang baxter equation quantum doubles of finite group algebras form a class of quasitriangular hopf algebras that algebraically solve the yang baxter equation each representation of the quantum double then gives a matrix solution of the yang baxter equation such solutions do not depend on a spectral parameter and to date there has been little investigation into extending these solutions such that they do depend on a spectral parameter here we first explicitly construct the matrix elements of the generators for all irreducible representations of quantum doubles of the dihedral groups dn these results may be used to determine constant solutions of the yang baxter equation we then discuss baxterization ansatze to obtain solutions of the yang baxter equation with a spectral parameter and give several examples including a new 21 vertex model we also describe this approach in terms of minimal dimensional representations of the quantum doubles of the alternating group a4 and the symmetric group s4

zamolodchikov s tetrahedron equation and hidden structure of quantum groups the tetrahedron equation is a three dimensional generalization of the yang baxter equation its solutions define integrable three dimensional lattice models of statistical mechanics and quantum field theory their integrability is not related to the size of the lattice therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices obviously any such three dimensional model can be viewed as a two dimensional integrable model on a square lattice where the additional third dimension is treated as an internal degree of freedom therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this hidden third dimension in this paper we construct a new solution of the tetrahedron equation which provides in this way the two dimensional solvable models related to finite dimensional highest weight representations for all quantum affine algebra where the rank n coincides with the size of the hidden dimension these models are related to an anisotropic deformation of the sl n invariant heisenberg magnets they were extensively studied for a long time but the hidden 3d structure was hitherto unknown our results lead to a remarkable exact rank size duality relation for the nested bethe ansatz solution for these models note also that the above solution of the tetrahedron equation arises in the quantization of the resonant three wave scattering model which is a well known integrable classical system in 2 1 dimensions

structure of regular representation of quantum universal enveloping algebra slq 2 with qp 1 the authors construct the regular representation quantum lie algebra of slq 2 and its quotient representations and then analyse the reducible structures of these representations when qp 1 for an integer p they show how the indecomposable representations of slq 2 are obtained by a purely algebraic method

diagonalisation of gl n invariant transfer matrices and quantum n wave system lee model the algebraic bethe ansatz is constructed for the gl n invariant transfer matrices with arbitrary gl n spin for this purpose the notion of vacuum subspace is introduced it is shown that the gl n magnet can be interpreted as an integrable discrete approximation of vector or matrix nonlinear schrodinger models or of the quantum n wave system

quantization of drinfel d doubles hopf algebra quantizations of four dimensional and six dimensional real classical drinfel d doubles are studied by following a direct analytic approach and the full quantization is explicitly obtained for most of them several new four and six dimensional quantum algebras are presented and some general features of the method are emphasized

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

universal baxterization for z graded hopf algebras we present a method for baxterizing solutions of the constant yang baxter equation associated with z graded hopf algebras to demonstrate the approach we provide examples for the taft algebras and the quantum group u q sl 2

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

a new braid like algebra for baxterisation we introduce a new baxterisation for r matrices that depend separately on two spectral parameters the baxterisation is based on a new algebra close to but different from the braid group we study representations of this new algebra on the vector space mathbb c m otimes n when the generators act locally the ones for m 2 are completely classified we also introduce some representations for generic m they allow us to recover the r matrix of the multi species generalization of the totally asymmetric simple exclusion process with different hopping rates

new solvable lattice models in three dimensions in this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models the first is the zamolodchikov model which is the only nontrivial model on a three dimen sional lattice so far solved the second is the chiral potts model on the square lattice and its generalization associated with theu q sl n algebra which is of current interest due to its connections with high genus algebraic curves and with representations of quantum groups at roots of unity we show that this last sl n generalized chiral potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square lattice layers with ann valued n 2 spin at each site further in then 2 case this three dimen sional model reduces after a modification of the boundary conditions to the zamolodchikov model we mentioned above

equivalent isotropic osp 2 2 spin chains and its bethe ansatz analysis in this paper we discuss the equivalence between two integrable and isotropic osp 2 2 chains via the braid monoid approach we use the analytical bethe ansatz approach in order to solve the spectrum of these systems the ground state is parametrized by a special structure of the bethe ansatz roots their interpretation in terms of correlated fermionic models is briefly discussed

asymptotically ordinary linear volterra difference equations with infinite delay abstract a class of linear volterra difference equations with infinite delay is considered it is shown that if the coefficient matrices are sufficiently small then the volterra difference equation is asymptotically equivalent to a linear ordinary difference equation at infinity the coefficient matrix of the ordinary difference equation is a solution of an associated matrix equation which can be obtained by successive approximations the eigenvalues of the approximating matrices converge exponentially to the characteristic roots of the volterra difference equation as a corollary an efficient new method is obtained for the computation of the characteristic roots with an explicit error estimate

an integral arising from the chiral sl n potts model we show that the integral can be expressed in terms of 5f4 hypergeometric functions the integral arises in the solution by baxter and bazhanov of the free energy of the sl n potts model which includes the term j 2 our result immediately gives the logarithmic mahler measure of the laurent polynomial in terms of the same hypergeometric functions