nonreduced extended affine weyl groups abstract in this paper we study the structure of the weyl groups of nonreduced extended affine root systems we show that similar to the case of reduced types an extended affine weyl group w of type bcl is semidirect product of a finite weyl group w of type bl and a heisenberg like normal subgroup h which is also a characteristic subgroup of w moreover h is of the form h h h 0 where both h and h 0 are normal subgroups of h with h h 0 1 h is naturally isomorphic to the root lattice of a finite root system of type bcl furthermore the semidirect product of w and h is isomorphic to the weyl group of a kac moody affine subroot system of r of type bcl

conjugacy classes in affine kac moody groups and principal g bundles over elliptic curves abstract for a simple complex lie group g the connected components of the moduli space of semistable g bundles over an elliptic curve are weighted projective spaces or quotients of weighted projective spaces by a finite group action in this note we will provide a new proof of this result using the invariant theory of affine kac moody groups in particular the action of the twisted coxeter element on the root system of g

integrable representations for generalized virasoro toroidal lie algebras abstract in this paper we give the structure of irreducible integrable modules of generalized virasoro toroidal lie algebras with finite dimensional weight spaces and non zero central charge

classification of irreducible integrable modules for toroidal lie algebras with finite dimensional weight spaces the purpose of this paper is to classify irreducible integrable modules for multi loop algebras let v be a vector space over the complex numbers let a t 1 t n be the laurent polynomials ring in n variables let va v a and let v m v t for m m1 mn n and t t1 1 t2 2 tn n let be a simple finite dimensional lie algebra over the complex numbers then a can be given a lie algebra structure see 1 1 let d1 dn be derivations defined by di x m mix m and let d be the linear span of d1 dn then a a is a lie algebra which we call a multi loop algebra the universal central externion of a is called a toroidal lie algebra and is studied in bb bc e3 e4 em so our multiloop algebras are quotients of toroidal lie algebras by centers fix a cartan subalgebra h of let be a graded homomorphism of u ha a see section 1 for details then we can define the universal highest weight module m for a let v be the unique irreducible quotient of m we then prove in theorem 3 4 that any irreducible integrable module for a is isomorphic to v by making use of a result from chari c we associate to a map from u ha by evaluating at 1 and an irreducible module v for a derivations do not act on v and therefore it cannot be extended to a now consider v a as a

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

exponential decay rate of solutions toward traveling waves for the cauchy problem of generalized benjamin bona mahony burgers equations abstract in this paper we investigate the exponential time decay rate of solutions toward traveling waves for the cauchy problem of generalized benjamin bona mahony burgers equations e u t u t x x u x x u x f u x 0 t 0 x r with prescribed initial data i u 0 x u 0 x u as x here 0 r are constants u are two given constants satisfying u u and the nonlinear function f u c 2 r is assumed to be either convex or concave based on the existence of traveling waves the local stability and the algebraic decay rate to traveling waves of solutions to the cauchy problem e and i established in yin et al 2007 13 we show an exponential decay rate of the solutions to the cauchy problem e and i toward the traveling waves mentioned above by employing the space time weighted energy method which was initiated by kawashima and matsumura in 1985 14 and later elaborated by matsumura and nishihara 1994 15 and nishikawa 1998 16

central extensions of current groups in two dimensions the paper is devoted to generalization of the theory of loop groups to the two dimensional case to every complex riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex lie group g by the jacobian of this surface this extension is topologically nontrivial as in the loop group case orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalg bundles over the surface when the surface is the torus elliptic curve classification of coadjoint orbits is related to linear difference equations in one variable and to classification of conjugacy classes in the loop group we study integral orbits for which the kirillov kostant form is a curvature form for some principal torus bundle the number of such orbits for a given level is finite as in the loop group case conjecturedly they correspond to analogues of integrable modules occurring in conformal field theory

partial classification of modules for lie algebra of diffeomorphisms of d dimensional torus we consider the lie algebra of the group of diffeomorphisms of a d dimensional torus which is also known to be the algebra of derivations on a laurent polynomial ring a in d commuting variables denoted by der a the universal central extension of der a for d 1 is the so called virasoro algebra the connection between virasoro algebra and physics is well known see for example the book on conformal field theory by di francesco mathieu and senechal in this paper we classify a der a modules which are irreducible and have finite dimensional weight spaces earlier larsson constructed a large class of modules the so called tensor fields based on gld modules which are also a modules we prove that they exhaust all a der a irreducible modules

convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions we prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions the result is applied to the discontinuous galerkin method due to cockburn hou and shu

extensions of some infinite algebras in space time in this paper the operator extensions for infinite algebras of residual gauge transformations and space time transformations in physical space time are discussed on general grounds possible general forms of the extensions are presented and their realizations in quantum field theories are discussed the meaning of nontriviality for the extensions will also be discussed in particular cohomology methods are used to calculate the extension of spatial diffeomorphism algebra when chiral fermions are coupled to both an external u 1 gauge field and gravity

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

on holomorphic factorization of wzw and coset models it is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the wzw model of two dimensional conformal field theory which previously have been understood primarily by using conformal field theory ward identities we also consider in a similar vein the holomorphic factorization ofg h coset models we discuss theg g model as a topological field theory and comment on a conjecture by spiegelglas

convergence of the finite volume method for multidimensional conservation laws we establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux functions our technique applies with a fairly unrestrictive assumption on the triangulations flat elements are allowed and to lipschitz continuous flux functions we treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of kruzkov the proof of convergence is based on a convergence framework coquel and lefloch math comp 57 1991 pp 169 210 and j numer anal 30 1993 pp 675 700 from a convex decomposition of the scheme we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities these estimates are shown to be sufficient for the passage to the limit in the discrete equation convergence follows from diperna s uniqueness result in the class of entropy measure valued solutions

four dimensional realization of two dimensional current groups the quotient realization of the central extensions of the current groups over riemann surfaces is achieved by means of the leray residue theory this approach replaces de rham cohomology in the classical wznw construction for affine lie groups

vertex representations for the ν 1 toroidal lie algebra of type bl abstract we give vertex operator constructions for the toroidal lie algebra of type b l 1 l 2 we prove that the subquotients of the modules are completely reducible and give the irreducible decomposition

a new algorithm based on lucas polynomials for approximate solution of 1d and 2d nonlinear generalized benjaminbonamahonyburgers equation in this paper a new method based on hybridization of lucas and fibonacci polynomials is developed for approximate solutions of 1d and 2d nonlinear generalized benjaminbonamahonyburgers equations firstly time discretization is made by using finite difference approaches after that unknown function and its derivatives are expanded to lucas series based on these series expansion differentiation matrices are derived by utilizing fibonacci polynomials by doing so the solution of the mentioned equations is reduced to the solution of an algebraic system of equations by solving this system of equations the lucas series coefficients are obtained then substituting these coefficients into lucas series expansion approximate solutions can be constructed successively the main goal of this paper is to indicate that lucas polynomial based method is appropriate for 1d and 2d nonlinear problems efficiency and performance of the proposed method are judged on six test problems which consists of the 1d and 2d version of mentioned equation by calculating l2 and l error norms feasibility of the method is verified by obtained accurate results

classification of irreducible bounded weight modules over the derivation lie algebras of quantum tori abstract let d 1 be an integer q q i j d d a d d complex matrix satisfying q i i 1 q i j q j i 1 with all q i j being roots of unity let c q be the rational quantum torus algebra associated with q and der c q its derivation lie algebra in this paper we give a complete classification of irreducible bounded weight modules over der c q they turn out to be irreducible sub quotients of der c q module v v w for a finite dimensional irreducible gl d module v a finite dimensional graded irreducible gl n module w and c d where the integer n is uniquely determined by q

geometry of the analytic loop group abstract we introduce and study a notion of analytic loop group with a riemann hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity u g with non trivial central charge we introduce a poisson structure and study properties of its poisson dual group we prove that the hopf poisson structure is isomorphic to the semi classical limit of the center of u g it is a geometric realization of the center then the symplectic leaves and corresponding equivalence classes of central characters of u g are parameterized by certain g bundles on an elliptic curve

looped cotangent virasoro algebra and non linear integrable systems in dimension 2 1 we consider a lie algebra generalizing the virasoro algebra to the case of two space variables we study its coadjoint representation and calculate the corresponding euler equations in particular we obtain a bi hamiltonian system that leads to an integrable non linear partial differential equation this equation is an analogue of the kadomtsev petviashvili of type b equation

quantum tori and the structure of elliptic quasi simple lie algebras abstract we study and classify those tame irreducible elliptic quasi simple lie algebras which are simply laced and of rankl 3 the first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates when the type isdorethe coordinates are laurent polynomials in variables while for typeathe coordinates can be any quantum torus in variables the next step is to study the universal central extension as well as the derivation algebra of the core these are related to the first connes cyclic homology group of the coordinates the final step is to use this information to give constructions of lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras

extended affine weyl groups abstract in this paper we study the weyl groups of reduced extended affine root systems the root systems of extended affine lie algebras we start by describing the extended affine weyl group as a semidirect product of a finite weyl group and a heisenberg like normal subgroup this provides a unique expression for the weyl group elements in terms of some naturally arisen transformations which is crucial in the further study of extended affine weyl groups we use this to give a presentation called a presentation by conjugation for an important subclass of extended affine weyl groups using a new notion called the index which is an invariant of the extended affine root systems we show that one of the important features of finite and affine root systems related to weyl group holds for the class of extended affine root systems

a new class of polynomial functions for approximate solution of generalized benjamin bona mahony burgers gbbmb equations abstract in this paper a new class of polynomial functions equipped with a parameter is used which is applied to approximate the gbbmb in this method the first time discretization of the gbbmb equations is made by using finite difference approaches then at any spatial point a nonlinear equations system is generated using the collocation points and solving a nonlinear system in the least squares method the coefficients of functions are obtained in some examples a linearization is performed first and then the collocation method is applied

algebras of diffeomorphisms of the n torus the general algebra d n v of infinitesimal diffeomorphisms of the n torus s1 n involving generators lv m depending on a structure vector v cn and a vector valued index m zn is constructed several results are proved for this algebra special cases of the algebra were previously presented in ramos shrock e ramos and r e shrock int j mod phys a 4 4279 1989 the concept of the q rank of the space v of structure vectors denoted rq v is defined and certain related cartan matrices are introduced it is shown that d n v as an ungraded algebra is simple if and only if rq v n i e the q rank of v is maximal a classification under isomorphisms is given for the algebra and is shown to reduce to a classification of the cartan matrices the space of structure vectors is isomorphic to the unique cartan subalgebra a number of properties concerning the central extensions of these algebras are then proved for the case dim v 1 rq v n it is shown that the central extension is unique fo

convergence of a staggered lax friedrichs scheme for nonlinear conservation laws on unstructured two dimensional grids based on nessyahu and tadmor s nonoscillatory central difference schemes for one dimensional hyperbolic conservation laws 16 for higher dimensions several finite volume extensions and numerical results on structured and unstructured grids have been presented the experiments show the wide applicability of these multidimensional schemes the theoretical arguments which support this are some maximum principles and a convergence proof in the scalar linear case a general proof of convergence as obtained for the original one dimensional nt schemes does not exist for any of the extensions to multidimensional nonlinear problems for the finite volume extension on two dimensional unstructured grids introduced by arminjon and viallon 3 4 we present a proof of convergence for the first order scheme in case of a nonlinear scalar hyperbolic conservation law

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

on the numerical treatment and analysis of benjamin bona mahony burgers equation in this paper the solution of benjamin bona mahony burgers bbmb equation is approximated by using collocation method stability of the method is discussed also the convergence of the method is proved the method is used on some examples and the numerical results have been obtained and compared with the exact solutions

highest weight modules over pre exp polynomial lie algebras abstract in this paper we introduce pre exp polynomial lie algebras which include loop algebras virasoro like algebras and some quantum torus lie algebras we study highest weight representations of these z n 1 graded lie algebras more precisely we show that non graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite dimensional weight spaces or not and they have all finite dimensional weight spaces if and only if the highest weight is an exp polynomial highest weight we also show that non graded and graded highest weight verma modules with the same highest weight are simultaneously irreducible or not and we determine necessary and sufficient conditions for a verma module to be irreducible

the numerical solution of nonlinear high dimensional generalized benjamin bona mahony burgers equation via the meshless method of radial basis functions in this paper a numerical technique is proposed for solving the nonlinear generalized benjamin bona mahony burgers equation firstly we obtain a time discrete scheme by approximating the first order time derivative via forward finite difference formula then we use kansa s approach to approximate the spatial derivatives we prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method also we show that convergence order of the time discrete scheme is o t we solve the two dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular triangular and circular domains and also the three dimensional case is solved on the cubical and spherical domains the aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations also several test problems including the three dimensional case are given numerical examples confirm the efficiency of the proposed scheme

error estimates for the staggered lax friedrichs scheme on unstructured grids staggered grid finite volume methods also called central schemes were introduced in one dimension by nessyahu and tadmor in 1990 in order to avoid the necessity of having information on solutions of riemann problems for the evaluation of numerical fluxes we consider the general case in multidimensions and on general staggered grids which have to satisfy only an overlap assumption we interpret the staggered lax friedrichs scheme as a three step method consisting of a prolongation step onto a finer intersection grid a finite volume step with an arbitrarily good numerical flux e g godunov flux on the intersection grid followed by an averaging step such that the calculation of numerical fluxes reduces to evaluations of the continuous flux using this point of view we prove an a posteriori error estimate and an a priori error estimate in the l1 norm in space and time which is of order h1 4 where h is a mesh size parameter hence we recover for the staggered lax friedrichs scheme the same order of convergence as for upwind finite volume methods on a fixed grid

kac moody groups topology of the dirac determinant bundle and fermionization the relation between kac moody groups and algebras and the determinant line bundle of the massless dirac operator in two dimensions is clarified analogous objects are studied in four space time dimensions and a generalization of witten s fermionization mechanism is presented in terms of the topology of the dirac determinant bundle

upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2d systems of nonlinear hyperbolic conservation laws in two space dimensions are considered which are characterized by the fact that the coupling of the equations is only due to source terms to solve these weakly coupled systems numerically a class of explicit and implicit upwind finite volume methods on unstructured grids is presented provided an unique entropy solution of the system of conservation laws exists we prove that the approximations obtained by these schemes converge for vanishing discretization parameter to this entropy solution these results are applied to examples from combustion theory and hydrology where the existence of entropy solutions can be shown the proofs rely on an extension of a result due to diperna concerning measure valued solutions to the case of weakly coupled hyperbolic systems

nonlinear stability of generalized benjamin bona mahony burgers shock profiles in several dimensions abstract this paper is concerned with nonlinear stability of planar viscous shock profiles of the generalized benjamin bona mahony burgers equations in two dimensions our analysis is motivated by j goodman s work 1989 4 on the nonlinear stability of scalar viscous shock profiles in two dimensions and is based on some new decay estimates on the planar viscous shock profiles of the generalized benjamin bona mahony burgers equations

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

irreducible representations of the lie algebra of the diffeomorphisms of ad dimensional torus in this paper we investigate the irreducibility of certain modules for the lie algebra of diffeomorphisms of torus t the lie algebra of diffeomorw 1 1 x phisms can be described as the derivations of a s c t t see 1 d w x w x rss we denote this lie algebra by der a larsson l1 constructed a functor from gl modules to der a modules in this paper we prove that d the image of a finite dimensional irreducible gl module is most often d irreducible the only exceptions are fundamental modules and the onedimensional modules in these cases we describe the sub modules and the quotients surprisingly the same class of modules image of finite dimen w x sional gl module are also constructed in e without any reference to d gl modules they are motivated by the vertex operator constructions in d w x em more details near the end of this introduction we will now describe our results in more detail let gl be the d lie algebras of d d matrices over complex numbers c let gl s sl d d c id where sl is the finite dimensional simple lie algebra of trace zero d matrices and id the identity matrix which is central it is well known that finite dimensional irreducible modules and dominant integral weights are in one one correspondence for a finite dimensional simple lie algebra so let v c be finite dimensional irreducible sl module corresponding to a d dominant integral weight c let id the central element act by a complex number b and denote the corresponding gl module by v c b d a d d larsson defined see 1 6 a functor f for a belonging to c rz from w x gl modules to der a modules in l1 these modules are weight modules d

integrable representations for the twisted full toroidal lie algebras the purpose of this work is to classify irreducible integrable modules of the twisted full toroidal lie algebra with finite dimensional weight spaces and non zero central charges there are three families of such modules two of which are classified

double coset construction of moduli space of holomorphic bundles and hitchin systems we present a description of the moduli space of holomorphic vector bundles over riemann curves as a double coset space which is differ from the standard loop group construction our approach is based on equivalent definitions of holomorphic bundles based on the transition maps or on the first order differential operators using this approach we present two independent derivations of the hitchin integrable systems we define a superfree upstairs system from which hitchin systems are obtained by three step hamiltonian reductions special attention is given to the schottky parameterization of curves

representations of twisted multi loop lie algebras abstract this paper describes finite dimensional irreducible representations of twisted multi loop lie algebras these representations are given in terms of the representations of finite dimensional semisimple lie algebras