local weyl modules for equivariant map algebras with free abelian group actions suppose a finite group acts on a scheme x and a finite dimensional lie algebra g the associated equivariant map algebra is the lie algebra of equivariant regular maps from x to g examples include generalized current algebras and twisted multiloop algebras local weyl modules play an important role in the theory of finite dimensional represen tations of loop algebras and quantum affine algebras in the current paper we extend the definition of local weyl modules previously defined only for generalized current algebras and twisted loop algebras to the setting of equivariant map algebras where g is semisimple x is affine of finite type and the group is abelian and acts freely on x we do so by defin ing twisting and untwisting functors which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues we also show that other properties of local weyl modules e g their characterization by homological prop erties and a tensor product property extend to the more general setting considered in the current paper

the restricted kirillov reshetikhin modules for the current and twisted current algebras we define a family of graded restricted modules for the polynomial current algebra associated to a simple lie algebra we study the graded character of these modules and show that they are the same as the graded characters of certain demazure modules in particular we see that the specialized characters are the same as those of the kirillov reshetikhin modules for quantum affine algebras

borel de siebenthal pairs global weyl modules and stanley reisner rings we develop the theory of integrable representations for an arbitrary standard maximal parabolic subalgebra of an affine lie algebra we see that such subalgebras can be thought of as arising in a natural way from a borel de siebenthal pair of semisimple lie algebras we see that although there are similarities with the representation theory of the standard maximal parabolic subalgebra there are also very interesting and non trivial differences including the fact that there are examples of non trivial global weyl modules which are irreducible and finite dimensional we also give a presentation of the endomorphism ring of the global weyl module although these are no longer polynomial algebras we see that for certain parabolics these algebras are stanley reisner rings which are both koszul and cohen macaulay

forced lattice vibrations part ii this is the first part of a two part series on forced lattice vibrations in which a semi infinite lattice of one dimensional particles xn n 1 is driven from one end by a particle x0 this particle undergoes a given periodically perturbed uniform motion x0 t at h yt where a and are constants and h has period 2 for a wide variety of restoring forces f i e f 0 numerical calculations indicate the existence of a sequence of thresholds 1 1 a h f 2 2 a h f k k a h f k 0 as k if k k 1 a k phase wave that is well described by the wave form emerges and travels through the lattice the goal of this series is to describe the emergence and calculate some properties of these wave forms in part i the authors first consider the case where f x ex i e toda forces but h is arbitrary and show how to compute a basic diagnostic see j formula 1 26 for the system in terms of the solution of an associated scalar riemann hilbert problem once a certain finite set of numbers is known in another direction the authors consider the case where f x is restoring but arbitrary and h is small here the authors prove a general result asserting that if there exists a sufficiently ample family of traveling wave solutions of the doubly infinite lattice then it is possible to construct time periodic k phase wave solutions with asymptotics in n of type iii for the driven system i in part ii the authors prove that sufficiently ample families of traveling wave solutions of the system iv exist in the cases 1 and 1 2 for general restoring forces f in the case with toda forces f x ex the authors prove that sufficiently ample families of traveling wave solutions

a spinorial formulation of the maximum clique problem of a graph we present a new formulation of the maximum clique problem of a graph in complex space we start observing that the adjacency matrix a of a graph can always be written in the form a b2 where b is a complex symmetric matrix formed by vectors of zero length null vectors and the maximum clique problem can be transformed in a geometrical problem for these vectors this problem in turn is translated in spinorial language and we show that each graph uniquely identifies a set of pure spinors that is vectors of the endomorphism space of clifford algebras and the maximum clique problem is formalized in this setting so that this much studied problem may take advantage from recent progresses of pure spinor geometry

on spinors and null vectors we investigate the relations between spinors and null vectors in clifford algebra of any dimension with particular emphasis on the conditions that a spinor must satisfy to be simple also pure in particular we prove i a new property for null vectors each of them bisects spinor space into two subspaces of equal size ii that simple spinors form one dimensional subspaces of spinor space iii a necessary and sufficient condition for a spinor to be simple that generalizes a theorem of cartan and chevalley which becomes a corollary of this result we also show how to write down easily the most general spinor with a given associated totally null plane

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

weyl demazure and fusion modules for the current algebra of sl r 1 we construct a poincare birkhoff witt type basis for the weyl modules v chari a pressley weyl modules for classical and quantum affine algebras represent theory 5 2001 191 223 math qa 0004174 of the current algebra of as a corollary we prove the conjecture made in v chari a pressley weyl modules for classical and quantum affine algebras represent theory 5 2001 191 223 math qa 0004174 v chari a pressley integrable and weyl modules for quantum affine in quantum groups and lie theory durham 1999 in london math soc lecture note ser vol 290 cambridge univ press cambridge 2001 pp 48 62 math qa 0007123 on the dimension of the weyl modules in this case further we relate the weyl modules to the fusion modules defined in b feigin s loktev on generalized kostka polynomials and the quantum verlinde rule in differential topology infinite dimensional lie algebras and applications in amer math soc transl ser 2 vol 194 1999 pp 61 79 math qa 9812093 of the current algebra and the demazure modules in level one representations of the corresponding affine algebra in particular this allows us to establish substantial cases of the conjectures in b feigin s loktev on generalized kostka polynomials and the quantum verlinde rule in differential topology infinite dimensional lie algebras and applications in amer math soc transl ser 2 vol 194 1999 pp 61 79 math qa 9812093 on the structure and graded character of the fusion modules

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

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 small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

fock space description of simple spinors cartan s simple often called pure spinors corresponding to even dimensional complex vector spaces are defined in terms of the associated maximal totally null planes their geometrical properties are derived and described using notions familiar to physicists dirac and weyl spinors gamma matrices tensors formed bilinearly from pairs of spinors and creation and annihilation operators of fermi states a new theorem characterizes a simple spinor by the properties of the vector t b where is an arbitrary spinor and b is the matrix connecting the gamma matrices with their transposes the cartan constraint equations on the components of simple spinors are given a new geometrically transparent derivation based on the action on simple spinors of a maximal abelian subgroup of the group spin

dispersive regularization of the whitham equation for the toda lattice in this paper the averaged whitham equations for the slow modulations of multi phase wavetrains for the toda lattice are studied our main result is showing how the solution with step initial conditions may be regularized by choosing an appropriate riemann surface on which the whitham equation is defined

weyl modules and q whittaker functions let g be a semi simple simply connected group over mathbb c following gerasimov et al comm math phys 294 97 119 2010 we use the q toda integrable system obtained by quantum group version of the kostant whittaker reduction cf etingof in am math soc trans ser 2 9 25 1999 sevostyanov in commun math phys 204 1 16 1999 to define the notion of q whittaker functions varpsi check lambda q z this is a family of invariant polynomials on the maximal torus t subset g here z in t depending on a dominant weight check lambda of g whose coefficients are rational functions in a variable q in mathbb c for a conjecturally the same but a priori different definition of the q toda system these functions were studied by ion duke math j 116 1 16 2003 and by cherednik int math res notices 20 3793 3842 2009 we shall denote the q whittaker functions from cherednik int math res notices 20 3793 3842 2009 by varpsi check lambda q z for g sl n these functions were extensively studied in gerasimov et al comm math phys 294 97 119 2010 comm math phys 294 121 143 2010 lett math phys 97 1 24 2011 we show that when g is simply laced the function hat varpsi check lambda q z varpsi check lambda q z cdot prod nolimits i in i prod nolimits r 1 langle alpha i check uplambda rangle 1 q r here i denotes the set of vertices of the dynkin diagram of g is equal to the character of a certain finite dimensional g mathsf t rtimes mathbb c module d check lambda the demazure module when g is not simply laced a twisted version of the above statement holds this result is known for varpsi check lambda replaced by varpsi check lambda cf sanderson in j algebraic combin 11 269 275 2000 and ion in duke math j 116 1 16 2003 however our proofs are algebro geometric and rely on our previous work braverman finkelberg in semi infinite schubert varieties and quantum k theory of flag manifolds arxiv 1111 2266 2011 and thus they are completely different from sanderson j algebraic combin 11 269 275 2000 and ion duke math j 116 1 16 2003 in particular we give an apparently new algebro geometric interpretation of the modules d check lambda

the soliton birth rate in the forced toda lattice the soliton birth rate in the semi infinite toda lattice is studied the lattice is forced by driving the zeroth particle with a constant velocity into the remainder of the lattice an approximate solution for the soliton birth rate is derived and it is shown to compare quite favorably with the actual birth rate

on the algorithmic complexity of associative algebras abstract the complexity l a of a finite dimensional associative algebra a is the number of non scalar multiplications divisions of an optimal algorithm to compute the product of two elements of the algebra we show l a 2ddim a t where t is the number of maximal two sided ideals of a

the forced toda lattice an example of an almost integrable system a method for solving forced integrable systems is presented the method requires the knowledge of at least one piece of information about the solution once this is known one may then construct the remainder of the solution in this sense these systems are almost integrable the forced semi infinite toda lattice is used as an example and to illustrate the method

on computational complexity of clifford algebra after a brief discussion of the computational complexity of clifford algebras we present a new basis for even clifford algebra cl 2m that simplifies greatly the actual calculations and without resorting to the conventional matrix isomorphism formulation obtains the same complexity in the last part we apply these results to the clifford algebra formulation of the np complete problem of the maximum clique of a graph introduced by budinich and budinich a spinorial formulation of the maximum clique problem of a graph j math phys 47 043502 2006

generalized weyl modules and nonsymmetric q whittaker functions abstract we introduce generalized global weyl modules and relate their graded characters to nonsymmetric macdonald polynomials and nonsymmetric q whittaker functions in particular we show that the series part of the nonsymmetric q whittaker function is a generating function for the graded characters of generalized global weyl modules