bilateral locally testable languages we give an algebraic characterization of a new variety of languages that will be called bilateral locally testable languages and denoted as blt given k 0 the membership of a word x to a blt k bt language can be decided by means of exploring the segments of length k of x as well as considering the order of appearance of those segments when we scan the prefixes and the suffixes of x in this paper we also characterize the syntactic semigroup of blt languages in terms of the equations of the variety they belong to as well as in terms of the join of two previously studied varieties

transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

optimal covering tours with turn costs we give the first algorithmic study of a class of covering tour problems related to the geometric traveling salesman problem find a polygonal tour for a cutter so that it sweeps out a specified region pocket in order to minimize a cost that depends mainly on the number of turns these problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems as well as arc routing postman problems with turn penalties we prove the np completeness of minimum turn milling and give efficient approximation algorithms for several natural versions of the problem including a polynomial time approximation scheme based on a novel adaptation of the m guillotine method

2 cocycles on the algebra of differential operators soit c t t 1 l anneau des polynomes de laurent sur un corps complexe c soit a c t t 1 d l algebre des operateurs differentiels sur c t t 1 soit 7b u l algebre des operateurs differentiels 7b u a gl u c sur c t t 1 gl u c tout 2 cocycle sur a est equivalent a un multiple de tout 2 cocycle sur 7b u est equivalent a un multiple de

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

framed torsion free sheaves on cp2 hilbert schemes and representations of infinite dimensional lie algebras abstract we construct representations of both heiseberg and clifford algebras on the equivariant cohomology of the moduli space of framed torsion free sheaves on p 2 as well as on the equivariant cohomology of the hilbert scheme of points on the resolution of a type a simple singularity this generalizes the earlier work of nakajima grojnowski and others

bosonic and fermionic realizations of the affine algebra g hat l_n we give an explicit description of all inequivalent heisenberg subalgebras of the affine lie algebra g hat l n mathbb c and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra the construction uses multicomponent fermionic fields and yields a correspondence between bosons elements of the heisenberg subalgebra and fermions

vertex operator representation of some quantum tori lie algebras we are defining the trigonometric lie subalgebras in bar x infty bar a infty bar b infty bar c infty bar d infty which are the natural generalization of the well known sin lie algebra the embedding formulas into bar x infty are introduced these algebras can be considered as some lie algebras of quantum tori an irreducible representation ofa b series of trigonometric lie algebras is constructed special cases of the trigonometric lie factor algebras which can be considered as a quantum preserving lie algebra structure deformation of the kac moody algebras are considered

on the coherence conjecture of pappas and rapoport we prove the generalized coherence conjecture proposed by pappas and rapoport as a corollary one of their theorems which describes the geometry of the special bers of the local models for ramied unitary groups holds unconditionally our proof is based on the study of the geometry in particular certain line bundles and adic sheaves of the global schubert varieties which are the equal characteristic counterparts of the local models

vertex operators symmetric functions and the spin group γn abstract this work provides a vertex operator approach to the symmetric group sn and its double covering group n by generalizing a result of frenkel and sato for sn we formulate a correspondence between the space v of certain twisted vertex operators the ring of symmetric functions over q 2 and the space of nontrivial irreducible characters of n under this identification we show that a distinguished orthogonal basis of v corresponds to the set of nontrivial irreducible characters of n where both are parametrized by partitions with odd integer parts the counterpart of this distinguished basis in the ring over q 2 is the set of schur s q functions which are loosely speaking the square roots of the schur functions the nontrivial part of the character table of n is shown to be given by certain matrix coefficients in v

on the minimum link length rectilinear spanning path problem complexity and algorithms the parameterized minimum link length rectilinear spanning path problem in the mbi d dimensional euclidean space mbi bbr d mbi d rsp for a given set mbi s of mbi n points in mbi bbr d and a positive integer mbi k is to find a piecewise linear path mbi p with at most mbi k line segments that covers i e contains all points in mbi s where all line segments in mbi p are axis parallel we first prove that the problem 2 rsp is np complete improving the previously known result that the problem 10 rsp is np complete we then consider a constrained mbi d rsp problem in which each line segment mbi s in the spanning path must cover all the points in the given set mbi s that share the same line with mbi s we present a new parameterized algorithm with running time mbi o ast 2d k for the constrained mbi d rsp problem which significantly improves the previous best result and is the first parameterized algorithm of running time mbi o ast 2 o k for the constrained mbi d rsp problem for a fixed mbi d we show that these results can be extended to the minimum link length rectilinear traveling salesman problem

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

a parameterized algorithm for the hyperplane cover problem we consider the problem of covering a given set of points in the euclidean space rm by a small number k of hyperplanes of dimensions bounded by d where d m we present a very simple parameterized algorithm for the problem and give thorough mathematical analysis to prove the correctness and derive the complexity of the algorithm when the algorithm is applied on the standard hyperplane cover problem in rd it runs in time o k d 1 k 1 3k improving the previous best algorithm of running time o kdk d for the problem when the algorithm is applied on the line cover problem in r2 it runs in time o kk 1 35k improving the previous best algorithm of running time o k2k 4 84k for the problem

learning in varieties of the form v li from positive data grammatical inference and finite semigroup theory are continuously developing nevertheless it seems that they are not interacting too much we propose in this paper an inference method for languages that belong to varieties of the form v li many well known families of languages like locally testable reversible dot depth one etc are of that form the method unifies existing algorithms for inference of some of those families and can be applied to some others that had not been inferred yet it uses a result about the cascade product in which one of the factors is a transducer and the second is the automaton obtained by inferring the base case of the family using the transduction of the sample as input

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

multiplicative structure in the stable splitting of ωsln c abstract the space of based loops in s l n c also known as the affine grassmannian of s l n c admits an e 2 or fusion product work of mitchell and richter proves that this based loop space stably splits as an infinite wedge sum we prove that the mitchell richter splitting is coherently multiplicative but not e 2 nonetheless we show that the splitting becomes e 2 after base change to complex cobordism our proof of the a splitting involves on the one hand an analysis of the multiplicative properties of weiss calculus and on the other a use of beilinson drinfeld grassmannians to verify a conjecture of mahowald and richter other results are obtained by explicit obstruction theoretic computations

level two standard an modules abstract in this paper we study the level two standard i e irreducible integrable highest weight modules for an affine special linear lie algebra of arbitrary rank in particular we give explicit realizations of these modules in the principal picture

symmetric function products and plethysms and the boson fermion correspondence we use the boson fermion correspondence for s and q functions to establish some interesting properties concerning outer products and plethysms of s functions or q functions by power sum symmetric functions the techniques which are developed are also applied to computing the inverse kostka foulkes matrix which is the transition matrix between hall littlewood symmetric functions and s functions in some simple cases

weight vectors of the basic a1 1 module and the littlewood richardson rule the basic representation of a1 1 is studied the weight vectors are represented in terms of schur functions a suitable base of any weight space is given the littlewood richardson rule appears in the linear relations among weight vectors

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

vector generation functions q spectral functions of hyperbolic geometry and vertex operators for quantum affine algebras we investigate the concept of q replicated argument in symmetric functions with its connection to spectral functions of hyperbolic geometry this construction suffices for vector generation functions in the form of q series and string theory we hope that the mathematical side of the construction can be enriched by ideas coming from physics

γ conformal algebras conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points we classify these algebras and their representations in terms of lie algebras and their representations with an action of the group to every conformal algebra and a character of we associate a lie algebra generated by fields with the ope with simple poles examples include twisted affine kac moody algebras the sin algebra which is a conformal analogue of the general linear algebra and its analogues the algebra of pseudodifferential operators on the circle etc

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

factorial representations of path groups abstract the reduction of the energy representation of the group of mappings from i 0 1 s11 r or r into a compact semisimple lie group g is given for g su 2 the factoriality of the representation which is of type iii in the case i r is proved

realization of the level two standard 2 1 modules in this paper we study the level two standard modules for the affine special linear lie algebras in particular we give the vertex operator realizations of all level two standard modules for the affine special linear lie algebras of odd rank

bosonization in background of metric and gauge the paper extends the two dimensional bosonization formulas on a riemann surface to the case with abelian gauge field the result is used to bosonize the massless thirring model and as an alternative route to the chiral bosonization

improved parameterized algorithms for minimum link length rectilinear spanning path problem the parameterized minimum link length rectilinear spanning path problem in the d dimensional euclidean space r d d rsp for a given set s of n points in r d and a positive integer k is to find a rectilinear spanning path p with at most k line segments that cover all points in s where all line segments in p are axis parallel in this paper we study a constrained d rsp problem constrained d rsp problem in which each line segment l in the spanning path must cover all the points in s that share the same line with l by applying the branch and search and dynamic programming techniques a parameterized algorithm with running time o 1 1 4 d 1 2 k is given for the constrained d rsp problem which significantly improves the current best result o 0 74 d k k

two constructions of affine lie algebra representations and boson fermion correspondence in quantum field theory we establish an isomorphism between the vertex and spinor representations of affine lie algebras for types dl 1 and dl 1 2 we also study decomposition of spinor representations using the infinite family of casimir operators and prove that they are either irreducible or have two irreducible components we show that the vertex and spinor constructions of the representations can be reformulated in the language of two dimensional quantum field theory in this physical context the two constructions yield the generalized sine gordon and thirring models respectively already in renormalized form the isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson fermion correspondence

twisted γ lie algebras and their vertex operator representations let be a generic subgroup of the multiplicative group c of nonzero complex numbers we define a class of lie algebras associated to called twisted lie algebras which are natural generalizations of the twisted affine lie algebras starting from an arbitrary even sublattice q of zn and an arbitrary finite order isometry of zn preserving q we construct a family of twisted vertex operators acting on generalized fock spaces which afford irreducible representations for certain twisted lie algebras as an application this recovers a number of known vertex operator realizations for infinite dimensional lie algebras such as twisted affine lie algebras extended affine lie algebras of type a trigonometric lie algebras of series a and b unitary lie algebras and bc graded lie algebras

vertex operator realization of symplectic and orthogonal s functions it is shown that symplectic and orthogonal schur functions can be realized by the action of the modes of certain vertex operators on the function the properties of these vertex operators allow one to reproduce the basic propreties of these types of symmetric functions the vertex operator modes are shown to obey free fermionic relations which provide applications such as calculating products and plethysms as well as generating hirota type partial differential equations pdes which have symplectic s functions as functions

annihilating fields of standard modules for affine lie algebras given an affine kac moody lie algebra tilde mathfrak g sigma of arbitrary type we determine certain minimal sets of annihilating fields of standard tilde mathfrak g sigma modules we then use these sets in order to obtain a characterization of standard tilde mathfrak g sigma modules in terms of irreducible loop tilde mathfrak g sigma modules which proves to be a useful tool for combinatorial constructions of bases for standard tilde mathfrak g sigma modules

dynamical systems on quantum tori lie algebras we use quantum tori lie algebras qtla which are a one parameter family of sub algebras ofgl to describe local and non local versions of the toda systems it turns out that the central charge of qtla is responsible for the non locality there are two regimes in the local systems conformal for irrational values of the parameter and non conformal and integrable for its rational values we also consider infinite dimensional analogs of rigid tops some of these systems give rise to quantized magneto hydrodynamic equations of an ideal fluid on a torus we also consider infinite dimensional versions of the integrable euler and clebsch cases

a sufficient condition to polynomially compute a minimum separating dfa the computation of a minimal separating automaton msa for regular languages has been studied from many different points of view from synthesis of automata or grammatical inference to the minimization of incompletely specified machines or compositional verification in the general case the problem is np complete but this drawback does not prevent the problem from having a real application in the above mentioned fields in this paper we propose a sufficient condition that guarantees that the computation of the msa can be carried out with polynomial time complexity

homology of affine springer fibers in the unramified case assuming a certain purity conjecture we derive a formula for the complex cohomology groups of the affine springer fiber corresponding to any unramified regular semisimple element we use this calculation to present a complex analog of the fundamental lemma for function fields we show that the kappa orbital integral that arises in the fundamental lemma is equal to the lefschetz trace of the frobenius acting on the etale cohomology of a related variety

realization of the level one standard _ 2 1 modules in this paper we study the level one standard or irreducible integrable highest weight modules for the affine symplectic lie algebras in particular we give concrete realizations of all level one standard modules for the affine symplectic lie algebras of even rank introduction in recent years explicit realizations of nontrivial representations of affine lie algebras have attracted the attention of many researchers because of their surprising connections with different areas of mathematics and physics among these realizations the principal realizations and the untwisted homogeneous realizations have proved to be especially useful in 1978 lepowsky and wilson 15 constructed the basic modules level one standard modules i e irreducible integrable highest weight modules for a in the principal realization in 1 o this construction was generalized to the level one standard modules for all simply laced affine lie algebras frenkel and kac 5 and segal 26 have given explicit construction of these modules in the untwisted realization the constructions of these modules in the general realization are given in 9 12 since then some higher level standard modules for certain simply laced affine lie algebras have also been constructed e g 14 17 18 19 21 22 25 however less is known regarding explicit constructions of standard modules for nonsimply laced affine lie algebras that is affine lie algebras of type b c f and g in the last section of 10 kac kazhdan lepowsky and wilson constructed reducible level one modules for the non simply laced affine lie algebras in the principal realization by taking fixed points of dynkin diagraminduced automorphisms recently in the untwisted realization the analogous representations again reducible have been constructed in 3 and 6 the level one standard modules for the affine orthogonal lie algebras b 1 have been n explicitly realized in 4 8 13 and 20 from different view points in 20 mandia has constructed the level one standard modules for the affine lie algebras f 1 and g 1 in 23 the author gave explicit constructions of the level one standard modules for the affine symplectic lie algebras of rank 3 and 4 received by the editors july 13 1989 presented to the 847th meeting of the ams pheonix arizona january 14 1989 1980 mathematics subject classification 1985 revision primary 17b65 1990 american mathematical society 0002 9947 90 1 00 25 per page

right and left locally testable languages the aim of this paper is to give a denition as well as an algebraic characterization of two new varieties of languages that will be referred to as right respectively left locally testable languages and denoted as rlt resp llt both families strictly contain the class of locally testable languages given k 0 the membership of a word x to a rlt k rt language can be decided by means of exploring the prex and sux of length k 1o fx and the segments of length k as well as considering the order of appearance of those segments when we scan the prexes of x membership of x to a k lt can be decided in a similar way but we have to change the word prexes for suxes in the above description in this paper we also show that s is a syntactic semigroup of a k rt resp k lt language if and only if the local subsemigroups of s are idempotents and right resp left repetition free c 2000 elsevier science b v all rights reserved

agt correspondence ding iohara algebra at roots of unity and lepowsky wilson construction it was recently conjectured that the agt correspondence between the instanton counting on and the two dimensional field theories with the conformal symmetry algebra can be considered as a root of unity limit of its k theoretic analogue from this point of view the algebra and a special basis in its representation are limits of the ding iohara algebra and the macdonald polynomials respectively in this paper we confirm this conjecture for the special case r 1 we uncover the implicit symmetry in this limit we also found that the vertex operators in the special basis have factorized aflt form

theta functions modular invariance and strings we use quillen s theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory in particular we show that the spin structure dependence of the chiral dirac determinant on a riemann surface is given by riemann s theta function we use this result to investigate the modular invariance of multiloop heterotic string amplitudes

the wronskian formalism for linear differential equations and pade approximations on demontre des theoremes generaux sur la determination des meilleures approximations rationnelles des solutions d equations differentielles lineaires a coefficients fonction rationnelle

ramanujan slater type identities related to the moduli 18 and 24 we present several new families of rogers ramanujan type identities related to the moduli 18 and 24 a few of the identities were found by either ramanujan slater or dyson but most are believed to be new for one of these families we discuss possible connections with lie algebras we also present two families of related false theta function identities

vertex operators arising from the homogeneous realization for we use the underlying fock space for the homogeneous vertex operator representation of the affine lie algebra to construct a family of vertex operators as an application an irreducible module for an extended affine lie algebra of type a n 1 coordinatized by a quantum torus q of 2 variables or 3 variables is obtained moreover this module turns out to be a highest weight module which is an analog of the basic module for affine lie algebras

vertex operator representation of the soliton tau functions in the an 1 toda models by dressing transformations we study the relation between the group algebraic approach and the dressing symmetry one to the soliton solutions of the an 1 toda field theory in 1 1 dimensions originally solitons in the affine toda models were found by olive turok and underwood single solitons are created by exponentials of elements which ad diagonalize the principal heisenberg subalgebra alternatively babelon and bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sine gordon model in this paper we show the equivalence between these two methods to construct solitons in the an 1 toda models

affine demazure modules and t fixed point subschemes in the affine grassmannian let g be a simple algebraic group defined over c and t be a maximal torus of g for a dominant coweight of g the t fixed point subscheme formula of the schubert variety formula in the affine grassmannian grg is a finite scheme we prove that for all such if g is of type a or d and for many of them if g is of type e there is a natural isomorphism between the dual of the level one affine demazure module corresponding to and the ring of functions twisted by certain line bundle on grg of formula we use this fact to give a geometrical proof of the frenkel kac segal isomorphism between basic representations of affine algebras of a d e type and lattice vertex algebras

over testable languages abstract in this paper we give a combinatorial description of the languages that are both locally and piecewise testable we call such languages over testable the locally testable semigroups were discovered in the study of finite automata the definition of locally testable semigroup is similar to that of locally testable language if we consider words over the alphabet which is the set of all elements of a semigroup s then such a word determines an element of s the product of the letters of the word a semigroup s is locally testable if whenever two words over the alphabet s have the same factors of a fixed length k and the same prefix and suffix of length k 1 then the products of the letters of these words are equal a natural extension of locally testable semigroups is obtained by dropping the condition about the prefix and suffix we call such semigroups strongly locally testable we prove in this paper that the family of strongly locally testable semigroups is exactly the intersection of the variety of locally idempotent and commutative semigroups and that of j trivial semigroups in particular a language is over testable if and only if its syntactic semigroup is strongly locally testable we use this algebraic characterization to derive a simple family of generators for the variety of over testable languages

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

additional symmetries of generalized integrable hierarchies the non isospectral symmetries of a general class of integrable hierarchies are found by generalizing the galilean and scaling symmetries of the korteweg de vries equation and its hierarchy the symmetries arise in a very natural way from the semi direct product structure of the virasoro algebra and the affine kac moody algebra underlying the construction of the hierarchy in particular the generators of the symmetries are shown to satisfy a subalgebra of the virasoro algebra when a tau function formalism is available the infinitesimal symmetries act directly on the tau functions as moments of virasoro currents some comments are made regarding the role of the non isospectral symmetries and the form of the string equations in matrix model formulations of quantum gravity in two dimensions and related systems

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

the t analog of the basic string function for twisted affine kac moody algebras abstract we study lusztig s t analog of weight multiplicities associated to level one representations of twisted affine kac moody algebras an explicit closed form expression is obtained for the corresponding t string function using constant term identities of macdonald and cherednik the closed form involves the generalized exponents of the graded pieces of the twisted affine algebra considered as modules for the underlying finite dimensional simple lie algebra this extends previous work on level 1 t string functions for the untwisted simply laced affine kac moody algebras

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

nonlocal integrable partners to generalized mkdv and two dimensional toda lattice equation in the formalism of a dressing method with quantized spectral parameter two new hierarchies milw2 and a two dimensional nonlocal toda lattice are constructed the characteristic property of the first one is the connection with the ilw2 hierarchy by means ofgl 2 miura transformation on the other hand milw2 equations turn out to be symmetry equations for a two dimensional nonlocal toda lattice a new version of the dressing technique with quantized spectral parameter is proposed