rational solution for the minimal representation of g2 the rational solution for the minimal representation of the quantum g2 is obtained as a limit of the trigonometric one presented by kuniba 1990

affine yangians and deformed double current algebras in type a we study the structure of yangians of affine type and deformed double current algebras which are deformations of the enveloping algebras of matrix w1 8 algebras we prove that they admit a pbw type basis establish a connection limit construction between these two types of algebras and toroidal quantum algebras and we give three equivalent definitions of deformed double current algebras we construct a schur weyl functor between these algebras and rational cherednik algebras

irreducible representations of yangians abstract we give explicit realizations of irreducible representations of the yangian of the general linear lie algebra and of its twisted analogues corresponding to symplectic and orthogonal lie algebras in particular we develop the fusion procedure for twisted yangians for the non twisted yangian this procedure goes back to the works of cherednik

quantum symmetric kac moody pairs the present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable kac moody algebras the resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras they give rise to triangular decompositions including a quantum analog of the iwasawa decomposition and they can be written explicitly in terms of generators and relations moreover their centers and their specializations are determined the constructions follow g letzter s theory of quantum symmetric pairs for semisimple lie algebras the main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable kac moody algebras due to kac and wang the resulting theory comprises various classes of examples which have previously appeared in the literature such as q onsager algebras and the twisted q yangians introduced by molev ragoucy and sorba

generalized bell states and principal realization of the yangian y sln we prove that the action of the yangian algebra y sln is better described by the principal generators on the tensor product of the fundamental representation and its dual the generalized bell states or maximally entangled states are permuted by the principal generators in a dramatically simple manner on the tensor product under the yangian symmetry the new quantum number j2 is also explicitly computed which gives an explanation for these maximally entangled states

coefficient identification in euler bernoulli equation from over posed data a method for solving the inverse problem for coefficient identification in the euler bernoulli equation from over posed data is presented the original inverse problem is replaced by a minimization problem the method is applied to the problem for identifying the coefficient in the case when it is a piece wise polynomial function several examples are elaborated and the numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the second order of approximation

r matrices and the magic square the distinguished representations associated with the rows of the freudenthal magic square have a uniform tensor product graph with edges labelled by linear functions of the dimension of the corresponding division algebra

rational k matrices and representations of twisted yangians we describe the twisted yangians y g h which arise as boundary remnants of yangians y g in 1 1d integrable field theories we describe and extend our recent construction of the intertwiners of their representations the rational boundary s or k matrices and perform a case by case analysis for all pairs g h giving the h decomposition of y g h representations where possible

an r matrix for d 3 4 this letter gives the calculation of a trigonometric r matrix for the twisted affine quantum group of type d 3 4

quantum twisted yangians symmetry baxterisation and centralizers based on the quantum twisted yangians integrable systems with special boundary conditions called soliton nonpreserving snp may be constructed in the present article we focus on the study of subalgebras of the quantum twisted yangians and we show that such a subalgebra provides an exact symmetry of the rational transfer matrix we discuss how the spectrum of a generic transfer matrix may be obtained by focusing only on two types of special boundaries it is also shown that the subalgebras emerging from the asymptotics of tensor product representations of the quantum twisted yangian turn out to be dual to the quantum brauer algebra to deal with general boundaries in the trigonometric case we propose a new algebra which also provides the appropriate framework for the baxterisation procedure in the snp case

contracted and expanded integrable structures we propose a generic framework to obtain certain types of contracted and centrally extended algebras this is based on the existence of quadratic algebras reflection algebras and twisted yangians naturally arising in the context of boundary integrable models a quite old misconception regarding the expansion of the e2 algebra into sl2 is resolved using the representation theory of the aforementioned quadratic algebras we also obtain centrally extended algebras associated with rational and trigonometric q deformed r matrices that are solutions of the yang baxter equation

connection between yangian symmetry and the quantum inverse scattering method the quantum nonlinear schrodinger model with two component fermions exhibits a yangian symmetry when considered on an infinite interval we construct the generators of the yangian using dunkl operators we show an equivalence between the monodromy matrix constructed from the dunkl operators and that of the quantum inverse scattering method under the yangian algebra the space of states with a fixed particle number forms a tensor product representation of fundamental representations

bosonic representations of yangian double with on the basis of the rtt ttr formalism we introduce the quantum double of the yangian for with a central extension the gauss decomposition of the t matrices gives us the so called drinfel d generators using these generators we present some examples of both finite and infinite dimensional representations that are quite natural deformations of their corresponding affine counterpart

quantum r matrix for g2 and a solvable 175 vertex model the quantum r matrix for cartan s exceptional simple lie algebra g2 in its seven dimensional minimal representation is presented the r matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra uq g2 this defines a new solvable seven state 175 vertex model on the planar square lattice whose boltzmann weights satisfy the yang baxter equation for the equivalent face model the local state probability is obtained in terms of a relative of a level 1 string function related to the affine lie algebra pair b3 1 contains implies g2 1

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

twisted yangians of small rank we study quantized enveloping algebras called twisted yangians associated with the symmetric pairs of types ci bdi and diii in cartan s classification when the rank is small we establish isomorphisms between these twisted yangians and the well known olshanskii s twisted yangians of types ai and aii and also with the molev ragoucy reflection algebras associated with symmetric pairs of type aiii we also construct isomorphisms with twisted yangians in drinfeld s original presentation

sklyanin determinant laplace operators and characteristic identities for classical lie algebras the structure of quantized enveloping algebras called twisted yangians which are naturally associated with the b c and d series of the classical lie algebras is studied an explicit formula for the formal series the sklyanin determinant whose coefficients are free generators of the center of the twisted yangian is obtained as a corollary a new system of algebraically independent generators of the center of the universal enveloping algebra for the orthogonal and symplectic lie algebras is obtained and the characteristic polynomial for the matrix formed by the generators of these lie algebras is found

irreducible finite dimensional representations of equivariant map algebras suppose a finite group acts on a scheme x and a finite dimensional lie algebra g the corresponding equivariant map algebra is the lie algebra m of equivariant regular maps from x to g we classify the irreducible finite dimensional representations of these algebras in particu lar we show that all such representations are tensor products of evaluation representations and one dimensional representations and we establish conditions ensuring that they are all evaluation representations for example this is always the case if m is perfect our results can be applied to multiloop algebras current algebras the onsager algebra and the tetrahedron algebra doing so we easily recover the known classifications of irreducible finite dimensional representations of these algebras moreover we obtain previously unknown classifica tions of irreducible finite dimensional representations of other types of equivariant map algebras such as the generalized onsager algebra

the embedding e0 and the spectrum dependent r matrix for q f4 the explicit embedding forms of e0 for the minimal representations of the quantum f4 e6 and e7 are obtained in this paper in terms of jimbo s method the spectrum dependent r matrices both trigonometric and rational for those representations are computed

twisted affine lie superalgebra of type q and quantization of its enveloping superalgebra we introduce a new quantum group which is a quantization of the enveloping superalgebra of a twisted affine lie superalgebra of type q we study generators and relations for superalgebras in the finite and twisted affine cases and also universal central extensions afterwards we apply the frt formalism to a certain solution of the quantum yang baxter equation to define that new quantum group and we study some of its properties we construct a functor of schur weyl type which connects it to affine hecke clifford algebras and prove that it provides an equivalence between two categories of modules

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

triality exceptional lie algebras and deligne dimension formulas abstract we give a computer free proof of the deligne cohen and de man formulas for the dimensions of the irreducible g modules appearing in g k k 4 where g ranges over the exceptional complex simple lie algebras we give additional dimension formulas for the exceptional series as well as uniform dimension formulas for other representations distinguished by freudenthal along the rows of his magic chart our proofs use the triality model of the magic square which we review and present a simplified proof of its validity we conclude with some general remarks about obtaining series of lie algebras in the spirit of deligne and vogel

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

isomorphism between the r matrix and drinfeld presentations of yangian in types b c and d it is well known that the gauss decomposition of the generator matrix in the r matrix presentation of the yangian in type a yields generators of its drinfeld presentation defining relations between these generators are known in an explicit form thus providing an isomorphism between the presentations it has been an open problem since the pioneering work of drinfeld to extend this result to the remaining types we give a solution for the classical types b c and d by constructing an explicit isomorphism between the r matrix and drinfeld presentations of the yangian it is based on an embedding theorem which allows us to consider the yangian of rank n 1 as a subalgebra of the yangian of rank n of the same type

a note on the existence and stability of an inverse problem for a sis model abstract in this note we discuss the existence and stability of an inverse problem arising from the determination of the reaction coefficients for an sis model the study is motivated by a remark regarding the final discussion of the recent paper by xiang and liu 2015 the weak assumption of the work of h xiang and b liu is that the proofs of existence and stability results are valid only for the one dimensional case here we introduce an appropriate framework which is also valid in the multidimensional case and that generalizes the previous results

rational representations of yangians associated with skew young diagrams consider the general linear group gl m over the complex field the irreducible rational representations of the group gl m can be labeled by the pairs of partitions and such that the total number of non zero parts of and does not exceed m let eq4 be the irreducible representation corresponding to such a pair regard the direct product as a subgroup of gl n m take any irreducible rational representation of gl n m the vector space comes with a natural action of the group gl n put n for any pair of standard young tableaux of skew shapes respectively we give a realization of as a subspace in the tensor product of n copies of defining representation of gl n and of n copies of the contragredient representation this subspace is determined as the image of a certain linear operator on w nnn we introduce this operator by an explicit multiplicative formula when m 0 and is an irreducible representation of gl n we recover the known realization of as a certain subspace in the space of all traceless tensors in then the operator may be regarded as the rational analogue of the young symmetrizer corresponding to the tableau of shape even when m 0 our formula for is new our results are applications of the representation theory of the yangian of the lie algebra in particular is an intertwining operator between certain representations of the algebra on we also introduce the notion of a rational representation of the yangian as a representation of the image of is rational and irreducible

local weyl modules and cyclicity of tensor products for yangians let mathfrak g be the exceptional complex simple lie algebra of type g2 we provide a concrete cyclicity condition for the tensor product of fundamental representations of the yangian y mathfrak g using this condition we show that every local weyl module is isomorphic to an ordered tensor product of fundamental representations of y mathfrak g

quantum spin chain with soliton non preserving boundary conditions we consider the case of an integrable quantum spin chain with soliton non preserving boundary conditions this is the first time that such boundary conditions have been considered in the spin chain framework we construct the transfer matrix of the model we study its symmetry and we find explicit expressions for its eigenvalues moreover we derive a new set of bethe ansatz equations by means of the analytical bethe ansatz method

inverse problem for coefficient identification in the euler bernoulli equation a special technique has been developed for the identification of the solution and the unknown coefficient in the euler bernoulli equation the original problem of the unknown coefficient identification from over posed data is transferred into a higher order well posed problem following the idea of the method of variational imbedding the new boundary value problem is solved by means of an iterative finite difference scheme the scheme is thoroughly validated and is shown to have a second order truncation error

weight bases of gelfand tsetlin type for representations of classical lie algebras this paper completes a series devoted to explicit constructions of finite dimensional irreducible representations of the classical lie algebras here the case of odd orthogonal lie algebras of type b is considered two previous papers dealt with c and d types a weight basis for each representation of the lie algebra 2n 1 is constructed the basis vectors are parametrized by gelfand tsetlin type patterns explicit formulae for the matrix elements of generators of 2n 1 in this basis are given the construction is based on the representation theory of the yangians

finite dimensional irreducible representations of twisted yangians we study quantized enveloping algebras called twisted yangians they are analogs of the yangian y gl n for the classical lie algebras of b c and d series the twisted yangians are subalgebras in y gl n and coideals with respect to the coproduct in y gl n we give a complete description of their finite dimensional irreducible representations every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite dimensional the result is analogous to drinfeld s theorem for the ordinary yangians its detailed proof for the a series is also reproduced for the simplest twisted yangians we construct an explicit realization for each finite dimensional irreducible representation in tensor products of representations of the corresponding lie algebras

representations of super yangian the classification of the finite dimensional irreducible representations of the super yangian associated with the lie superalgebra gl 1 1 is presented in detail

yangian of the queer lie superalgebra consider the complex matrix lie superalgebra with the standard generators where define an involutory automorphism of by the twisted polynomial current lie superalgebra has a natural lie co superalgebra structure we quantise the universal enveloping algebra as a co poisson hopf superalgebra for the quantised algebra we give a description of the centre and construct the double in the sense of drinfeld we also construct a wide class of irreducible representations of the quantised algebra

rational r matrices in irreducible representations working directly from the yang baxter equation the author obtains the spectral decomposition of rational r matrices in irreducible representations together with a necessary condition for their existence examples are given and connections with the drinfeld yangian construction are discussed

an inverse problem for estimation of bending stiffness in kirchhoff love plates this work deals with the development of a numerical method for solving an inverse problem for bending stiffness estimation in a kirchhoff love plate from overdetermined data the coefficient is identified using a technique called the method of variational imbedding where the original inverse problem is replaced by a minimization problem the euler lagrange equations for minimization comprise higher order equations for the solution of the displacement and an equation for the bending stiffness the correctness of the embedded problem is discussed a difference scheme and a numerical algorithm for solving the parameter identification problem are developed numerical results for the obtained values of the bending stiffness as an inverse problem are presented

boundary scattering symmetric spaces and the principal chiral model on the half line we investigate integrable boundary conditions bcs for the principal chiral model on the half line and rational solutions of the boundary yang baxter equation bybe in each case we find a connection with type i riemannian globally symmetric spaces g h there is a class of integrable bcs in which the boundary field is restricted to lie in a coset of h these bcs are parametrized by g h g h there are rational solutions of the bybe in the defining representations of all classical g parametrized by g h and using these we propose boundary s matrices for the principal chiral model parametrized by g h g h which correspond to our boundary conditions

new boundary conditions for integrable lattices new boundary conditions for classical integrable nonlinear lattices of the xxx type such as the heisenberg chain and the toda lattice are presented these integrable extensions are formulated in terms of a generic xxx heisenberg magnet interacting with two additional spins at each end of the chain the construction uses the most general rank 1 ansatz for the 2 2 l operator satisfying the reflection equation algebra with rational r matrix the associated quadratic algebra is shown to be that of dynamical symmetry for the a1 and bc2 calogero moser problems other physical realizations of our quadratic algebra are also considered

twisted super yangians and their representations starting with the super yangian y m n based on gl m n we define twisted super yangians y m n only y m 2n and y 2m n can be defined and appear to be isomorphic one with each other we study their finite dimensional irreducible highest weight representations

hopf structure of the yangian y sln in the drinfel d realization the yangian of the lie algebra sln is known to have different presentations in particular the rtt realization and the drinfel d realization using the isomorphism between them the explicit expressions of the comultiplication the antipode and the counit in the drinfel d realization of the yangian y sln are given as examples the cases of y sl2 and y sl3 are worked out

bulk and boundary g2 factorized s matrices we investigate the g2 invariant bulk 1 1d factorized s matrix constructed by ogievetsky using the bootstrap on the three point coupling of the vector multiplet to constrain its cdd ambiguity we then construct the corresponding boundary s matrix demonstrating it to be consistent with y g2 a1 a1 symmetry

twisted yangians twisted quantum loop algebras and affine hecke algebras of type bc we study twisted yangians of type aiii which have appeared in the literature under the name of reflection algebras they admit q versions which are new twisted quantum loop algebras we explain how these can be defined equivalently either via the reflection equation or as coideal subalgebras of yangians of gln resp of quantum loop algebras of gln the connection with affine hecke algebras of type bc comes from a functor of schur weyl type between their module categories

symmetric pairs for quantized enveloping algebras abstract let be an involution of a semisimple lie algebra g let g denote the fixed lie subalgebra and assume the cartan subalgebra of g has been chosen in a suitable way we construct a quantum analog of u g which can be characterized as the unique subalgebra of the quantized enveloping algebra of g which is a maximal right coideal that specializes to u g

hidden yangians in 2d massive current algebras we define non local conserved currents in massive current algebras in two dimensions our approach is algebraic and non perturbative the non local currents give a quantum field realization of the yangians we show how the noncocommutativity of the yangians is related to the non locality of the currents we discuss the implications of the existence of non local conserved charges on thes matrices

on the existence of a global minimum in inverse parameters identification by self optimizing inverse analysis method abstract in this paper a mathematical proof of the existence of a global minimum of self optim self optimizing inverse analysis method cost functional is presented based upon weak solution theory of partial differential equations the self optim provides single global minimum rather than having multiple global minima corresponding to unrealistic solutions of the inverse problem furthermore discrete approximation of the inverse problem and computational methods for the cost functional are proposed and the proof is numerically verified this paper provides a rigorous mathematical foundation for applications of the self optim method to various inverse problems in mechanics

twisted yangians for symmetric pairs of types b c d we study a class of quantized enveloping algebras called twisted yangians associated with the symmetric pairs of types b c d in cartan s classification these algebras can be regarded as coideal subalgebras of the yangian for orthogonal or symplectic lie algebras they can also be presented as quotients of a reflection algebra by additional symmetry relations we prove an analogue of the poincare birkoff witt theorem determine their centres and study also extended reflection algebras

the full set of cn invariant factorized s matrices the author uses the method of the tensor product graph to construct rational yangian invariant solutions of the yang baxter equation in fundamental representations of cn and thence the full set of cn invariant factorized s matrices

boundary k matrices for the six vertex and the n 2n 1 an 1 vertex models boundary conditions compatible with integrability are obtained for two dimensional models by solving the factorizability equations for the reflection matrices k or theta for the six vertex model the general solution depending on four arbitrary parameters is found for the an 1 models all diagonal solutions are found the associated integrable magnetic hamiltonians are explicitly derived

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

an inverse problem for the plate in the love kirchhoff theory this paper studies an inverse boundary value problem for the equation of the flexure of the linear inhomogeneous isotropic thin plate in the context of the love kirchhoff theory it is shown that the young modulus e and the poisson ratio nu of the material forming the plate can be determined by the dirichlet to neumann map the boundary values of e nu and their finitely many derivatives if e and nu are close enough to constants in a suitable sense

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

inverse problem for coefficient identification in sir epidemic models abstract this work deals with the development of a numerical method for solving an inverse problem for identifying coefficients from over posed data in an sir mathematical model of infectious diseases spread through a population the parameters are identified using a generalized least squares method which is similar to the technique called method of variational imbedding where the original inverse problem is replaced by a minimization problem a difference scheme and a numerical algorithm for solving the parameter identification problem are developed the correctness of the embedded problem is discussed numerical results of the parameters representing the solution to the inverse problem are presented

classical nonlinear σ models on grassmann manifolds of compact or noncompact type grassmannian models are reexamined in the light of a new geometrical result namely the cartan immersion of a riemannian symmetric space g h into its isometry group g is not always a diffeomorphism onto the set m introduced by eichenherr and forger nucl phys b 164 528 1980 this is the case in particular for grassmann manifolds of all types their set m is shown to be disconnected and its structure is completely analyzed as a consequence a new constraint must be introduced in the backlund transformation bt method proposed by harnad saint aubin and shnider commun math phys 92 329 1984 it is shown however to always be satisfied for grassmann manifolds of compact type i e g compact but the problem remains open in most other cases on the other hand the bt method is extended to the euclidean regime

on irreducibility of tensor products of evaluation modules for the quantum affine algebra every irreducible finite dimensional representation of the quantized enveloping algebra can be extended to the quantum affine algebra via the evaluation homomorphism we give in explicit form the necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules

quantum r matrix for e7 and f4 groups the quantum r matrix for e7 and f4 are explicitly constructed for the defining representations 56 and 26 of e7 and f4 respectively for these skein relations link polynomials and spectral parameter dependent solutions of yang baxter equation are obtained the r matrix for the matrix elements related by the weyl reflection has different values surprisingly

the theory of ends pushdown automata and second order logic abstract a class of edge labeled graphs called context free are defined according to their behavior at infinity such graphs are generalizations of cayley graphs of context free groups they are also shown to be definable in a very natural way in terms of push down automata using rabin s theorem on the monadic second order theory of the finite binary tree these graphs are also shown to have a decidable monadic second order theory questions about tiling systems and cellular automata operating on these graphs are decidable even when the analogous questions are not when the systems operate on a two dimensional grid

comment on solutions of the yang baxter equation for isotropic quantum spin chains we comment on a recent paper by kennedy j phys a math gen 25 1992 2809 in which a systematic search for integrable spin s su 2 invariant quantum chains for s or 6 revealed four spin s families of integrable chains along with an additional integrable chain at s 3 we identify these su 2 invariant chains with known g invariant r matrices where g is a simple lie algebra and give arguments that kennedy s results may well constitute the complete classification of integrable spin s su 2 invariant chains

quantum group symmetries and non local currents in 2d qft we construct and study the implications of some new non local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions including the sine gordon theory and its generalization to affine toda theory these non local currents provide a non perturbative formulation of the theories the symmetry algebras correspond to the quantum affine kac moody algebras thes matrices are completely characterized by these symmetries formals matrices for the imaginary coupling affine toda theories are thereby derived the application of theses matrices to perturbed coset conformal field theory is studied non local charges generating the finite dimensional quantum group in the liouville theory are briefly presented the formalism based on non local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d

bethe subalgebras in twisted yangians we study analogues of the yangian of the lie algebra mathfrak g mathfrak l n for the other classical lie algebras mathfrak s mathfrak o n and mathfrak s mathfrak p n we call them twisted yangians they are coideal subalgebras in the yangian of mathfrak g mathfrak l n and admit homomorphisms onto the universal enveloping algebras u mathfrak s mathfrak o n and u mathfrak s mathfrak p n respectively in every twisted yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical lie algebra the images in u mathfrak s mathfrak o n and u mathfrak s mathfrak p n of these subalgebras are also maximal commutative

rational r matrices centralizer algebras and tensor identities for e6 and e7 exceptional families of lie algebras we use cvitanovic s group theory princeton university press princeton nj in press http www nbi dk grouptheory phys rev d 14 1536 1976 diagrammatic techniques to construct the rational solutions of the yang baxter equation yang baxter equation in integrable systems edited by m jimbo advanced series in mathematical physics vol 10 world scientific singapore 1990 associated with the e6 and e7 families of lie algebras and thus explain westbury s j phys a 36 2857 2003 observations about their uniform spectral decompositions in doing so we explore the extensions of the brauer and symmetric group algebras to the centralizer algebras of e7 and e6 on their lowest dimensional representations and up to threefold tensor products thereof giving bases for them and a range of identities satisfied by the algebras defining invariant tensors

rtt presentation of finite algebras we construct a wide class of finite algebras as truncations of yangians these truncations correspond to algebra homomorphisms and allow us to construct the algebras as exchange algebras the r matrix being the yangian one as an application we classify all irreducible finite dimensional representations of these algebras and determine their centre

rational and polynomial representations of yangians abstract we define natural classes of rational and polynomial representations of the yangian of the general linear lie algebra we also present the classification and explicit realizations of all irreducible rational representations of the yangian

drinfeld functor and finite dimensional representations of yangian we extend the results of drinfeld on the drinfeld functor to the case l n we present the character of finite dimensional representations of the yangian y mathfrak s mathfrak l n in terms of the kazhdan lusztig polynomials as a consequence

representation theory of rectangular finite w algebras abstract we classify the finite dimensional irreducible representations of rectangular finite w algebras i e the finite w algebras u g e where g is a symplectic or orthogonal lie algebra and e g is a nilpotent element with jordan blocks all the same size

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

representations of the exceptional and other lie algebras with integral eigenvalues of the casimir operator the uniformity for the family of exceptional lie algebras g o f the decompositions of the powers of their adjoint representations is now well known for powers up to four the paper describes an extension of this uniformity for the totally antisymmetrized nth powers up to n 9 identifying see tables 3 and 6 f amilies of representations with integer eigenvalues 5 9 for the quadratic casimir operator in each case providing a formula see equations 11 15 for the dimensions of the representations in the family as af unction of d dim g t his generalizes previous results for powers j and casimir eigenvalues j j 4 many intriguing perhaps puzzling features of the dimension formulae are discussed and the possibility that they may be valid for a wider class of not necessarily simple lie algebras is considered

solutions to the quantum yang baxter equation for the exceptional lie algebras with a spectral parameter the trigonometric solutions to the quantum yang baxter equation with a spectral parameter for the fundamental representations of the exceptional lie algebras e6 e7 and e8 will be explicity presented moreover reduction and inversion like formulas are obtained in the presence of the spectral parameter

the large scale geometry of strongly aperiodic subshifts of finite type abstract a subshift on a group g is a closed g invariant subset of a g for some finite set a it is said to be a subshift of finite type sft if it is defined by a finite collection of forbidden patterns to be strongly aperiodic if all point stabilizers are trivial and weakly aperiodic if all point stabilizers are infinite index in g we show that groups with at least 2 ends have a strongly aperiodic sft and that having such an sft is a qi invariant for finitely presented groups we show that a finitely presented torsion free group with no weakly aperiodic sft must be qi rigid the domino problem on g asks whether the sft specified by a given set of forbidden patterns is empty we show that decidability of the domino problem is a qi invariant

universal characters from the macdonald identities abstract the main result of kostant invent math 158 1 2004 181 226 arxiv math gr 0309232 shows that a sequence of complexes associated with the macdonald identities gives a sequence of universal characters which extends and generalises some representations of the exceptional series given in deligne c r acad sci paris ser math 322 4 1996 321 326 this proves the conjectures in macfarlane pfeiffer j phys a 36 2003 230 5 2317 arxiv math ph 0208014 in this paper we use a twisted version of this definition to define a sequence of complexes for the first row of the freudenthal magic square we also calculate the dimensions of these characters for the special linear groups and for the symplectic groups this gives two new explicit expressions for the d arcais polynomials

r matrix presentation for super yangians y osp m 2n we give a rtt presentation of super yangians y g for g osp m 2n thereby unifying the formalism with the cases of g so n and g sp 2n

the orthogonal eigenbasis and norms of eigenvectors in the spin calogero sutherland model using a technique based on the yangian gelfand zetlin algebra and the associated yangian gelfand zetlin bases we construct an orthogonal basis of eigenvectors in the calogero sutherland model with spin and derive product type formulae for norms of these eigenvectors

quantum superalgebras at roots of unity and non abelian symmetries of integrable models we consider integrable vertex models whose boltzmann weights r matrices are trigonometric solutions to the graded yang baxter equation as is well known the latter can be generically constructed from quantum affine superalgebras uq these algebras do not form a symmetry algebra of the model for generic values of the deformation parameter q when periodic boundary conditions are imposed if q is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non abelian subalgebras which are translation invariant and commute with the transfer matrix and therefore with all charges of the model in the line of argument we introduce the restricted quantum superalgebra uresq and investigate its root of unity limit we prove several new formulae involving supercommutators of arbitrary powers of the chevalley serre generators and derive higher order quantum serre relations as well as an analogue of lustzig s quantum frobenius theorem for superalgebras