symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

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

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

on zamolodchikov s solution of the tetrahedron equations the tetrahedron equations arise in field theory as the condition for thes matrix in 2 1 dimensions to be factorizable and in statistical mechanics as the condition that the transfer matrices of three dimensional models commute zamolodchikov has proposed what appear from numerical evidence and special cases to be non trivial particular solutions of these equations but has not fully verified them here it is proved that they are indeed solutions

pairwise compatibility graphs a survey a graph g v e is a pairwise compatibility graph pcg if there exists an edge weighted tree t and two nonnegative real numbers d min and d max such that each leaf u of t is a node of v and there is an edge u v in e if and only if d min leq d t u v leq d max where d t u v is the sum of weights of the edges on the unique path from u to v in t in this article we survey the state of the art concerning this class of graphs and some of its subclasses

quantum 2 1 evolution model a quantum evolution model in 2 1 discrete spacetime connected with a 3d fundamental map is investigated map is derived as a map providing a zero curvature of a 2d linear lattice system called the current system in a special case of the local weyl algebra for dynamical variables the map appears to be canonical and it corresponds to the known operator valued matrix the current system is a type of the linear problem for the 2 1 evolution model a generating function for the integrals of motion for the evolution is derived with the help of the current system thus the complete integrability in 3d is proved directly

composition of kinetic momenta the sl 2 casecase the tensor products of restricted and unrestricted finite dimensional irreducible representations of open image in new window sl 2 are considered forq a root of unity they are decomposed into direct sums of irreducible and or indecomposable representations

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

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

a note on the generalized lie algebra in a recent paper dobrev and sudbery classified the highest weight and lowest weight finite dimensional irreducible representations of the quantum lie algebra introduced by lyubashenko and sudbery the aim of this note is to add to this classification all the finite dimensional irreducible representations which have no highest weight and or no lowest weight in the case when q is a root of unity for this purpose we give a description of the enlarged centre

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

double coset decompositions of reductive lie groups arising from two involutions g 0 h g and g 0 l g here g g g g g for an automorphism of g and f0 denote the connected component of f containing the identity e for a lie group f in 8 we gave standard representatives of double coset decompositions h g l for some typical triples g h l by only using algebraic method as the most elementary example we studied the case g h l gl n if gl p if gl n p if gl r if gl n r if for an arbitrary field if we also studied in 8 other examples related to quadratic forms on vector spaces over the fields ir c and ih in this paper we assume that g is a reductive lie group and we will describe the structure of the double coset decomposition h g l for an arbitrary g h l first we study the compact case section 3 and next the noncompact case section 4 in section 3 we assume that the semisimple part gs of g is compact gs is the analytic subgroup of g for gs g g when g gs and

some constant solutions to zamolodchikov s tetrahedron equations the author presents constant solutions to the tetrahedron equations proposed by zamolodchikov 1981 in general from a given solution of the yang baxter equation there are two ways to construct solutions to the tetrahedron equation there are also other kinds of solutions they present some two dimensional solutions that were obtained by directly solving the equations using either an upper triangular or zamolodchikov s ansatz

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

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 ansatz for the three layer zamolodchikov model this paper is a continuation of our previous work boos h e and mangazeev v v 1999 j phys a math gen 32 3041 54 we obtain two more functional relations for the eigenvalues of the transfer matrices for the sl 3 chiral potts model at q2 1 this model up to a modification of boundary conditions is equivalent to the three layer three dimensional zamolodchikov model from these relations we derive the bethe ansatz equations

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

mim width iii graph powers and generalized distance domination problems abstract we generalize the family of problems and locally checkable vertex partition problems to their distance versions which naturally captures well known problems such as distance r dominating set and distance r independent set we show that these distance problems are in xp parameterized by the structural parameter mim width and hence polynomial time solvable on graph classes where mim width is bounded and quickly computable such as k trapezoid graphs dilworth k graphs circular permutation graphs interval graphs and their complements convex graphs and their complements k polygon graphs circular arc graphs complements of d degenerate graphs and h graphs if given an h representation we obtain these results by showing that taking any power of a graph never increases its mim width by more than a factor of two to supplement these findings we show that many classes of problems are w 1 hard parameterized by mim width solution size we show that powers of graphs of tree width w 1 or path width w and powers of graphs of clique width w have mim width at most w these results provide new classes of bounded mim width we prove a slight strengthening of the first statement which implies that surprisingly leaf power graphs which are of importance in the field of phylogenetic studies have mim width at most 1

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

theta function parametrization and fusion for 3d integrable boltzmann weights we report progress in constructing boltzmann weights for integrable threedimensional lattice spin models we show that a large class of vertex solutions to the modified tetrahedron equation mte can be conveniently parametrized in terms of nth roots of theta functions on the jacobian of a compact algebraic curve fay s identity guarantees the fermat relations and the classical equations of motion for the parameters determining the boltzmann weights our parametrization allows us to write a simple formula for fused boltzmann weights r which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation imposing periodic boundary conditions we observe that the r satisfy the normal tetrahedron equation the scheme described contains the zamolodchikov baxter bazhanov model and the chessboard model as special cases

fast dynamic programming for locally checkable vertex subset and vertex partitioning problems given a graph g we provide dynamic programming algorithms for many locally checkable vertex subset and vertex partitioning problems their runtime is polynomial in the number of equivalence classes of problem specific equivalence relations on subsets of vertices defined on a given decomposition tree of g using these algorithms all these problems become solvable in polynomial time for many well known graph classes like interval graphs and permutation graphs belmonte and vatshelle 2013 1 given a decomposition of boolean width k we show that the algorithms will have runtime o n 42 o k 2 providing the first large class of problems solvable in fixed parameter single exponential time in boolean width

hyperbolic coxeter groups for triangular potts models the symmetry groups generated by the inversion relations of lattice models of statistical mechanics on triangular lattices are analysed for vertex models and for the standard scalar potts model with two and three site interactions these groups are generated by three inversion relations and are seen to be generically very large ones hyperbolic groups two situations for which the representations of these groups degenerate into smaller ones hopefully compatible with integrability are considered the first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left or right hand side of a yang baxter relation in this case the representation of the group is isomorphic up to a semi direct product by a finite group to z z the second reduction for q state ports models occurs for particular values of q the so called tutte beraha numbers for this model algebraic varieties including the known ferromagnetic critical variety happen to be invariant under such large groups of symmetries as a byproduct this analysis provides nice birational representations of hyperbolic coxeter groups

invariant domains in the complexification of a noncompact riemannian symmetric space let g k be a noncompact riemannian symmetric space and let gc kc be its complexification then g acts on gc kc by left translations we study the invariant cr structure of the closed g orbits of maximal dimension and determine which ones can lie in the boundary of an invariant stein domain in this way we obtain information on the g invariant stein domains in gc kc

two state spectral free solutions of the frenkel moore simplex equation whilst many solutions have been found for the quantum yang baxter equation qybe there are fewer known solutions available for its higher dimensional generalizations zamolodchikov s tetrahedron equation zte and frenkel and moore s simplex equation fme in this paper we present families of solutions to the fme which may help us to understand more about higher dimensional generalization of the qybe

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

an integrable 3d lattice model with positive boltzmann weights in this paper we construct a three dimensional 3d solvable lattice model with non negative boltzmann weights the spin variables in the model are assigned to edges of the 3d cubic lattice and run over an infinite number of discrete states the boltzmann weights satisfy the tetrahedron equation which is a 3d generalization of the yang baxter equation the weights depend on a free parameter 0 q 1 and three continuous field variables the layer to layer transfer matrices of the model form a two parameter commutative family this is the first example of a non trivial solvable 3d lattice model with non negative boltzmann weights

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

baxter equations and the zamolodchikov model abstract a summary is given of the significance of the star triangle or yang baxter equations in the solution of two dimensional statistical mechanical models the extension to the three dimensional tetrahedron relations is quoted and it is shown how this can be used to obtain the partition function of the zamolodchikov model for an n lattice the n result differs from that recently obtained by bazhanov and stroganov

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

on the parameterized complexity of multiple interval graph problems multiple interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it many applications of interval graphs also generalize to multiple interval graphs often allowing for more robustness in the modeling of the specific application with this motivation in mind a recent systematic study of optimization problems in multiple interval graphs was initiated in this sequel we study multiple interval graph problems from the perspective of parameterized complexity the problems under consideration are k independent set k dominating set and k clique which are all known to be w 1 hard for general graphs and np complete for multiple interval graphs we prove that k clique is in fpt while k independent set and k dominating set are both w 1 hard we also prove that k independent dominating set a hybrid of the two above problems is also w 1 hard our hardness results hold even when each vertex is associated with at most two intervals and all intervals have unit length furthermore as an interesting byproduct of our hardness results we develop a useful technique for showing w 1 hardness via a reduction from the k multicolored clique problem a variant of k clique we believe this technique has interest in its own right as it should help in simplifying w 1 hardness results which are notoriously hard to construct and technically tedious

induced module construction for highest weight representations of uq gl n at roots of unity a method is investigated for inducing highest weight representations for the quantum group uq gl n from the canonical subalgebra uq gl n 1 when q is a root of unity we classify the irreps into two types typical and atypical where the former is a generalization of the class of irreps with maximal dimensionality the structures of both the typical and atypical irreps are studied in particular a sufficiency condition is given for an irrep to be typical as examples we consider flat representations induced from a one dimensional representation of the canonical subalgebra and representations induced from vector representation

remarks on the quantum dilogarithm a quantum analogue of the dilogarithm function has been introduced by faddeev and kashaev 1994 in such a way that a certain identity in the weyl algebra wq plays the role of the five term dilogarithm identity we study this identity in the limit when q approaches a root of unity and show that it then reduces to the restricted star triangle relation which has been used previously by bazhanov and baxter 1992 as a local integrability condition of a class of three dimensional solvable lattice models

a labelling scheme for higher dimensional simplex equations we present a succinct way of obtaining all possible higher dimensional generalizations of the quantum yang baxter equation qybe using the scheme we could generate the two popular three simplex equations namely the zamolodchikov tetrahedron equation and the frenkel and moore equation

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

three dimensional vertex model in statistical mechanics from baxter bazhanov model we find that the boltzmann weight of the three dimensional baxter bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the wu kadanoff wegner duality between the cube and vertex type tetrahedron equations is obtained explicitly for the baxter bazhanov model then a three dimensional vertex model is obtained by considering the symmetry property of the weight function which corresponds to the three dimensional baxter bazhanov model the vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube we write down the symmetry relations of the weight functions under the actions of the symmetry groupg of the cube the six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected with the six spaces in which the vertextype tetrahedron equation is defined

partition function of a three dimensional solvable model abstract the solvable generalized 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 with the specific two and three spin interactions on the body centered cubic lattice here we report the results of a calculation of the partition function per site for the infinite lattice

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

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

functional relations and nested bethe ansatz 3 chiral potts model at q 2 1 we obtain the functional relations for the eigenvalues of the transfer matrix of the sl 3 chiral potts model for for the homogeneous model in both directions a solution of these functional relations can be written in terms of roots of bethe ansatz like equations in addition a direct nested bethe ansatz has also been developed for this case

dilogarithm identities in conformal field theory and group homology recently rogers dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations one of the connecting threads is an identity of richmond szekeres that appeared in the computation of central charges in conformal field theory we show that the richmond szekeres identity and its extension by kirillov reshetikhin equivalent to an identity found earlier by lewin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2 2 real matrices viewed as adiscrete group this connection allows us to clarify a few of the assertions and conjectures stated in the work of nahm recknagel terhoven concerning the role of algebraic k theory and thurston s program on hyperbolic 3 manifolds specifically it is not related to hyperbolic 3 manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2 2 real matrices viewed as a topological group this also resolves the weaker version of the conjecture as formulated by kirillov we end with a summary of a number of open conjectures on the mathematical side

quasitriangularity of quantum groups at roots of 1 an important property of a hopf algebra is its quasitriangularity and it is useful for various applications this property is investigated for quantum groupssl2 at roots of 1 it is shown that different forms of the quantum groupsl2 at roots of 1 are either quasitriangular or have similar structure which will be called braiding in the most interesting cases this property means that braiding automorphism is a combination of some poisson transformation and an adjoint transformation with a certain element of the tensor square of the algebra

visible actions on flag varieties of exceptional groups and a generalization of the cartan decomposition abstract we give a generalization of the cartan decomposition for connected compact exceptional lie groups motivated by the work on visible actions of t kobayashi t kobayashi j math soc japan 59 2007 669 691 for type a groups this paper extends his results to the exceptional groups first we classify a pair of levi subgroups l h of any compact exceptional simple lie group g such that g l g h where is a chevalley weyl involution this implies that the natural l action on the generalized flag variety g h is strongly visible and likewise the h action on g l and the g action on g g l h are strongly visible second we find a generalized cartan decomposition g l b h with b in g by using the herringbone stitch method which was introduced by kobayashi applications to multiplicity free representations are also discussed

hardness of computing width parameters based on branch decompositions over the vertex set many width parameters of graphs are defined using branch decompositions over the vertex set of the graph and a corresponding cut function in this paper we give a general framework for showing hardness of many width parameters defined in such a way by reducing from the problem of deciding the exact value of the cut function we show that this implies np hardness for deciding both boolean width and mim width and that mim width is w1 hard and not in apx unless np zpp

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

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

double coset decompositions of algebraic groups arising from two involutions i let g be a real reductive algebraic group with two involutive automorphisms involutions and 2 id put h g and l g where g x g x x for an automorphism of g if g is connected and one of the following three conditions is satisfied then the structure of the double coset decomposition h g l is studied a g is compact and b or is a cartan involution c in b we may assume that 1 9 if a or b is satisfied then we can choose representatives of h g l in a maximal connected abelian subgroup a contained in g g g g g 1 4 5 9 consider the case c by the map g g g 1 the double coset decomposition h g h is identified with the decomposition of g into h cojugacy classes it is easy to see that the set g is closed under the jordan decomposition let x xsxu be the jordan decomposition of x g then xs xu g c f 14 the structure of semisimple elements in g is studied in 14 by generalizing the notion of cartan subgroups let q be the subspace of the lie algebra g of g defined by q x g x x then the exponential map gives a bijection between the nilpotent h conjugacy classes in q and the unipotent h conjugacy classes in g nilpotent h conjugacy classes in q and their singularities closure relations etc are studied in several papers 7 12 13 11 16 and 17 especially suppose that g g1 g1 and x y y x for x y g1 then h g x x x g1 since g h is identified with g1 by the map x y h 7 xy 1 the double coset decomposition h g h is identified with the decomposition of g1 into conjugacy classes

a solvable lattice model and related rogers ramanujan type identities an exactly solvable lattice model is presented which is a natural generalization of baxter s hard hexagon model to three spin states the free energy and spin densities are calculated related rogers ramanujan type identities are also studied

the emerging role of number theory in exactly solvable models in lattice statistical mechanics abstract we review the baxter model the hard hexagon model and their multistate generalizations from a point of view that stresses the connection with modular functions and additive number theory it is shown for example that various physical quantities in the hard hexagon model are all expressible in terms of modular functions the use of rogers ramanujan type identities in solvable models is also reviewed

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

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

on solving the ising model functional relation without elliptic functions the exactly solvable n state chiral potts model differs from previous solvable lattice models in that its mathematics leads to hyperelliptic rather than ordinary elliptic functions it is still an open question whether these hyperelliptic functions provide a useful computational tool in particular it has recently been shown that the functional relation for the eigenvalues of the transfer matrix can be solved directly without introducing such functions when n 2 the model reduces to the ising model and this method provides a novel and straightforward way of solving the ising model in fact the method simplifies considerably here we present the resulting technique

the six and eight vertex models revisited elliott lieb s ice type models opened up the whole field of solvable models in statistical mechanics here we discuss the commuting transfer matrix t qequations for these models writing them in a more explicit and transparent notation that we believe offers new insights the approach manifests the relationship between the six vertex and chiral potts models and between the eight vertex and kashiwara miwa models

periodic and partially periodic representations of su n q the gelfand zetlin basis is adapted tosu n q forq a root of unit extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center a crucial identity is derived and proved which guarantees the periodicity of the action of the generators full periodicity is relaxed by stages some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation in the extreme case of highest weight representations all the raising and lowering operators are nilpotent as an alternative approach an auxiliary algebra giving all the periodic representations is presented an explicit solution of this system forn 3 while fully equivalent to the g z basis turns out to be much simpler

discrete series representations for the orbit spaces arising from two involutions of real reductive lie groups leth gbe real reductive lie groups a discrete series representation for a homogeneous spaceg his an irreducible representation ofgrealized as a closedg invariant subspace ofl2 g h the condition for the existence of discrete series representations forg hwas not known in general except for reductive symmetric spaces this paper offers a sufficient condition for the existence of discrete series representations forg hin the setting thatg his a homogeneous submanifold of a symmetric space g hwhereg g h we prove that discrete series representations are non empty for a number of non symmetric homogeneous spaces such assp 2n r sp n0 c gl n1 c gl nk c nj n ando 4m n u 2m j 0 2j n

integrability from 2d n 2 2 dualities we study integrable models in the context of the recently discovered gauge ybe correspondence where the yang baxter equation is promoted to a duality between two su persymmetric gauge theories we study flavored elliptic genus of 2d n 2 2 quiver gauge theories which theories are defined from statistical lattices regarded as quiver diagrams our r matrices are written in terms of theta functions and simplifies considerably when the gauge groups at the quiver nodes are abelian we also discuss the modularity properties of the r matrix reduction of 2d index to 1d witten index and string theory realizations of our theories

duality between the trigonometric bcn sutherland system and a completed rational ruijsenaars schneider van diejen system we present a new case of duality between integrable many body systems where two systems live on the action angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one our investigation utilizes an idea that was exploited previously to provide group theoretic interpretation for several dualities discovered originally by ruijsenaars in the group theoretic framework one applies hamiltonian reduction to two abelian poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space taking the cotangent bundle of u 2n as the upstairs space we demonstrate how this mechanism leads to a new dual pair involving the bcn trigonometric sutherland system thereby we generalize earlier results pertaining to the an trigonometric sutherland system as well as a recent work by pusztai on the hyperbolic bcn sutherland system

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

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

orbits and invariants associated with a pair of commuting involutions let be commuting involutions of the connected reductive algebraic group g where and g are defined over a usually algebraically closed field k char k 2 we have fixed point groups h g and k g and an action h k g g where h k g hgk 1 h h k k g g let g h k denote spec o g h k the categorical quotient let a be maximal among subtori s of g such that s s s 1 for all s s there is the associated weyl group w wh k a we show the inclusion a g induces an isomorphism a w g h k in particular the closed h k orbits are precisely those which intersect a the fibers of g g h k are the same as those occurring in certain associated symmetric varieties in particular the fibers consist of finitely many orbits we investigate the structure of w and its relation to other naturally occurring weyl groups and to the action of on the a weight spaces of the relation of the orbit type stratifications of a w and g h k along the way we simplify some of richardson s proofs for the symmetric case and at the end we quickly recover results of berger flensted jensen hoogenboom and matsuki ber57 fj78 hoo84 mat97 for the case k

representations of uq sl n at roots of unity the gelfand zetlin basis for representations of uq sl n is improved to better fit the case when q is a root of unity the usual q deformed representations as well as the nilpotent periodic cyclic semi periodic semi cyclic and also some atypical representations are now described with the same formalism

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

solutions of the quantum yang baxter equation with extra nonadditive parameters we present a systematic technique to construct solutions to the yang baxter equation which 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 we exploit the fact that quantum non compact algebras such as uq su 1 1 and type i quantum superalgebras such as uq gl 1 mod 1 and uq gl 2 mod 1 are known to admit non trivial one parameter families of infinite dimensional and finite dimensional irreps respectively even for generic q we develop a technique for constructing the corresponding spectral dependent r matrices as examples we work out the the r matrices for the three quantum algebras mentioned above in certain representations

diagonalization of the transfer matrix of a nonintersecting string model abstract in this paper we obtain the eigenvectors of the transfer matrix of a nonintersecting string model using the bethe ansatz the result is in agreement with baxter s conjecture of weak equivalence of the present model with the six vertex model

graph classes with structured neighborhoods and algorithmic applications given a graph in any of the following graph classes trapezoid graphs circular permutation graphs convex graphs dilworth k graphs k polygon graphs circular arc graphs and complements of k degenerate graphs we show how to compute decompositions with the number of d neighborhoods bounded by a polynomial of the input size combined with results of bui xuan telle and vatshelle 2013 1 this leads to polynomial time algorithms for a large class of locally checkable vertex subset and vertex partitioning problems on all of these graph classes the boolean width of a graph is related to the number of 1 neighborhoods and our results imply that any of these graph classes have boolean width o logn

roots of unity representations of quantum groups representations of quantum groupsu e g n g n any semi simple lie algebra of rankn are constructed from arbitrary representations of rankn 1 quantum groups for e a root of unity representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases

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

statistical mechanics a perspective on the work of r j baxter abstract the work of rodney baxter in statistical mechanics is surveyed with an emphasis on the role that his exact calculations have played in the development of the modern theory of critical phenomena

real double coset spaces and their invariants abstract let g be a real form of a complex reductive group suppose that we are given involutions and of g let h g denote the fixed group of and let k g denote the fixed group of we are interested in calculating the double coset space h g k we use moment map and invariant theoretic techniques to calculate the double cosets especially the ones that are closed one salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles

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

quantum group sigma models field theoretic models for fields taking values in quantum groups are investigated first the authors consider the suq 2 sigma model q real expressed in terms of basic notions of noncommutative differential geometry they discuss the case in which the sigma model fields are represented as products of conventional sigma fields and of the coordinate independent algebra an explicit example is provided by the uq 2 sigma model with qn 1 in which case quantum matrices uq 2 are realized as 2n 2n unitary matrices open problems are pointed out

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

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

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

equivalence of domains arising from duality of orbits on flag manifolds iii in gindikin and matsuki 2003 we defined a invariant subset of for each orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the akhiezer gindikin domain if is of nonholomorphic type this conjecture was proved for closed in wolf and zierau 2000 and 2003 fels and huckleberry 2005 and matsuki 2006 and for open in matsuki 2006 it was proved for the other orbits in matsuki 2006 when is of non hermitian type in this paper we prove the conjecture for an arbitrary non closed orbit when is of hermitian type thus the conjecture is completely solved affirmatively

classification of admissible nilpotent orbits in the classical real lie algebras m duflo introduced the notion of admissibility of a linear form on a real lie algebra and constructed certain unitary representations from admissible linear forms with good polarization in d d vogan has wanted to find a notion of unipotent representations of real reductive groups which corresponds to the one of lusztig s unipotent representations of finite reductive groups he pointed out that a unipotent representation should be a constituent of a representation attached to an admissible nilpotent coadjoint orbit by some orbit method cf vl v2 but for real reductive groups there have not yet been found the correct definition of unipotent representations and hence the orbit method which should exist i e a correspondence between admissible nilpotent coadjoint orbits and representations which contain unipotent representations as constituents however it seems that these notions must play an important role for the classification of unitary representations the purpose of this paper is to determine all the admissible nilpotent orbits in real classical non compact lie groups here a real classical lie group means a lie group in the following list rg

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

at roots of unity the quantum group at roots of unity is introduced by means of duality pairings with the quantum algebra its irreducible representations corresponding to the type of the quantum algebra are constructed through the universal t matrix the irreducible representations corresponding to the type are also found an invariant integral on this quantum group is given endowed with that some properties like unitarity and orthogonality of the irreducible representations are discussed

automorphisms of algebraic varieties and yang baxter equations it is shown that commuting transfer matrix models in statistical mechanics are parametrized by algebraic varieties having a set of automorphisms deduced from the so called inversion relation in general this set of automorphisms is infinite this shows that for algebraic varieties of dimension 1 the models are parametrized by algebraic curves of genus 0 or 1