q difference equation and q polynomials we establish one general q exponential operator we give one general q difference equation relationship between q exponential operator and q difference equation are given using q difference equation to set up a q polynomials hn some generating functions for hn are obtained in this paper we establish one general q exponential operator identity by solving one simple q difference equation using this q difference equation some multilinear generating functions for h n are discussed

the symmetry and exact solutions of the nonlinear d alembert equation for complex fields the nonlinear wave equations for the complex scalar field invariant under a conformal group are constructed and multiparametrical exact solutions of certain nonlinear complex d alembert equations are found

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

continuous subgroups of the generalized schrodinger groups some general results on the subalgebras of the lie algebra asch n of the generalized schrodinger group sch n and on the subalgebras of the lie algebra asch n of the generalized extended schrodinger group sch n have been obtained the subalgebra structure of the algebras asch n and asch n are studied with respect to inner automorphisms of the groups sch n and sch n respectively the maximal abelian subalgebras and the one dimensional subalgebras of the algebras asch n and asch n have been explicitly found the full classification of the subalgebras of the algebras asch 3 asch 3 which are nonconjugate to the subalgebras of asch 2 asch 2 respectively has been carried out

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

a remark on andrews askey integral in this paper we use the andrews askey integral and the q chu vandermonde formula to derive a more general integral formula applications of the new integral formula are also given which include to derive the q pfaff saalschutz formula and the terminating sears s 23 transformation formula

note on a q contour integral formula abstract we derive a more general askey and wilson q contour integral formula in this paper from this general integral formula and using the symmetry of the integral formula we can conclude some general sears terminating 4 3 transformation formula

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

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

on the reduction of the nonlinear multi dimensional wave equations and compatibility of the d alembert hamilton system abstract the necessary conditions of the compatibility of the d alembert hamilton system in minkowsky space r 1 n are established the problem of reduction of p 1 n invariant wave equations to ordinary differential equations is discussed

tau functions and generalized integrable hierarchies the tau function formalism for a class of generalized zero curvature integrable hierarchies of partial differential equations is constructed the class includes the drinfel d sokolov hierarchies a direct relation between the variables of the zero curvature formalism and the tau functions is established the formalism also clarifies the connection between the zero curvature hierarchies and the hirotatype hierarchies of kac and wakimoto

reduction and exact solutions of the navier stokes equations the authors construct a complete set of g 1 3 inequivalent ansatze of codimension 1 for the navier stokes ns field which reduce the ns equations to systems of ordinary differential equations ode having solved these odes they thereby obtain solutions of the ns equations formulae of group multiplication of solutions are given several non lie ansatze are discussed

on the classification of subalgebras of the conformal algebra with respect to inner automorphisms we give a complete justification of the classification of inequivalent subalgebras of the conformal algebra with respect to the inner automorphisms of the conformal group and we perform the classification of the subalgebras of the conformal algebra ac 1 3

new scale invariant nonlinear differential equations for a complex scalar field abstract we describe all complex wave equations of the form u f u u invariant under the extended poincare group as a result we have obtained the five new classes of p 1 3 invariant nonlinear partial differential equations for the complex scalar field

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

new conditional symmetries and exact solutions of the nonlinear wave equation we have constructed the new class of conditional symmetries of the nonlinear wave equation in 1 3 dimensions these symmetries enable us to obtain broad families of new exact solutions of the nonlinear wave equation containing up to the four arbitrary functions

computation of generalized equivariant cohomologies of kac moody flag varieties abstract in 1998 goresky kottwitz and macpherson showed that for certain projective varieties x equipped with an algebraic action of a complex torus t the equivariant cohomology ring h t x can be described by combinatorial data obtained from its orbit decomposition in this paper we generalize their theorem in three different ways first our group g need not be a torus second our space x is an equivariant stratified space along with some additional hypotheses on the attaching maps third and most important we allow for generalized equivariant cohomology theories e g instead of h t for these spaces we give a combinatorial description of e g x as a subring of e g f i where the f i are certain invariant subspaces of x our main examples are the flag varieties g p of kac moody groups g with the action of the torus of g in this context the f i are the t fixed points and e g is a t equivariant complex oriented cohomology theory such as h t k t or mu t we detail several explicit examples

amplitude phase representation for solutions of nonlinear d alembert equations we consider the nonlinear complex d alembert equation square operator psi f mod psi mod psi with psi represented in terms of amplitude and phase in 1 n dimensional minkowski space we exploit a compatible d alembert hamilton system to construct new types of exact solutions for some nonlinearities

covering algebras ii isomorphism of loop algebras this paper studies the loop algebras that arise from pairs consisting of a symmetrizable kac moody lie algebra g and a finite order automorphism of g we obtain necessary and sufficient conditions for two such loop algebras to be isomorphic

two dimensional representations of affine kac moody groups abstract we give a complete and explicit description for all two dimensional representations of non twisted affine kac moody groups those representations are found having a unique form characterized by homomorphisms between the base fields the homomorphisms from affine kac moody groups to two dimensional linear groups are also determined

on subalgebras of the lie algebra of the extended poincare group p 1 n some general results on the subalgebras of the lie algebra ap 1 n of the extended poincare group p 1 n n 2 with respect to p 1 n conjugation have been obtained all subalgebras of ap 1 4 that are nonconjugate to the subalgebras of ap 1 4 are classified with respect to p 1 4 conjugation the list of representatives of each conjugacy class is presented

on cartan subalgebras abstract in this paper we study the concept of a cartan subalgebra in the context of locally finite and borcherds kac moody lie algebras

kac moody groups and integrability of soliton equations a new approach to integrability of affine toda field theories and closely related to them kdv hierarchies is proposed the flows of a hierarchy are explicity identified with infinitesimal action of the principal abelian subalgebra of the corresponding affine kac moody algebra on a homogeneous space

on a reduction and solutions of nonlinear wave equations with broken symmetry a generalised definition for invariance of partial differential equations is proposed exact solutions of the equations with broken symmetry are obtained

the solution space of the unitary matrix model string equation and the sato grassmannian the space of all solutions to the string equation of the symmetric unitary one matrix model is determined it is shown that the string equation is equivalent to simple conditions on pointsv1 andv2 in the big cell gr 0 of the sato grassmannian gr this is a consequence of a well defined continuum limit in which the string equation has the simple form open image in new window matrices of differential operators these conditions onv1 andv2 yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation this geometric formulation leads directly to the virasoro constraintsln n 0 whereln annihilate the two modified kdv functions whose product gives the partition function of the unitary matrix model

continuous subgroups of the poincare group p 1 4 an exhaustive description of the nonsplitting subalgebras of the lp 1 4 algebra with respect to p 1 4 conjugation is presented

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

special conformal groups of a riemannian manifold and lie point symmetries of the nonlinear poisson equation abstract we obtain a complete group classification of the lie point symmetries of nonlinear poisson equations on generic pseudo riemannian manifolds m using this result we study their noether symmetries and establish the respective conservation laws it is shown that the projection of the lie point symmetries on m are special subgroups of the conformal group of m in particular if the scalar curvature of m vanishes the projection on m of the lie point symmetry group of the poisson equation with critical nonlinearity is the conformal group of the manifold we illustrate our results by applying them to the thurston geometries

a complete basis set of functionally independent invariants of continuous lie groups using the methods of differential equations the author has obtained the complete set of functionally independent bases of invariants for some continuous lie groups of mathematical physics i the conformal group ii the semi direct product of the weyl group and the one parameter group of psi translation iii the direct product of the poincare group and the one parameter group of psi scaling iv the weyl group v the 14 parameter maximal symmetry group of the schrodinger equation for the coulombic system of atoms and molecules and vi su 3 of elementary particle physics the first four groups are the only groups which together with the poincare group are the maximal symmetry groups of all forms of the non linear klein gordon equation

cartan subalgebras of root reductive lie algebras abstract root reductive lie algebras are direct limits of finite dimensional reductive lie algebras under injections which preserve the root spaces it is known that a root reductive lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of finite dimensional simple lie algebras as well as copies of the three simple infinite dimensional root reductive lie algebras sl so and sp as part of a structure theory program for root reductive lie algebras cartan subalgebras of the lie algebra gl were introduced and studied in k h neeb i penkov cartan subalgebras of gl canad math bull 46 2003 597 616 in the present paper we refine and extend the results of k h neeb i penkov cartan subalgebras of gl canad math bull 46 2003 597 616 to the case of a general root reductive lie algebra g we prove that the cartan subalgebras of g are the centralizers of maximal toral subalgebras and that they are nilpotent and self normalizing we also give an explicit description of all cartan subalgebras of the simple lie algebras sl so and sp we conclude the paper with a characterization of the set of conjugacy classes of cartan subalgebras of the lie algebras gl sl so and sp with respect to the group of automorphisms of the natural representation which preserve the lie algebra

orthogonal and non orthogonal separation of variables in the wave equation utt uxx v x u 0utt uxx v x u 0 we develop a direct approach to the separation of variables in partial differential equations within the framework of this approach the problem of the separation of variables in the wave equation with time independent potential reduces to solving an over determined system of nonlinear differential equations we have succeeded in constructing its general solution and as a result all potentials v x permitting variable separation have been found for each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study it should be noted that the above approach yields both orthogonal and non orthogonal systems of coordinates

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

preliminary group classification of a class of fourth order evolution equations we perform preliminary group classification of a class of fourth order evolution equations in one spatial variable following the approach developed by basarab horwath et al acta appl math 69 43 2001 we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple lie groups in addition we describe all fourth order evolution equations from the class under consideration which are invariant under solvable lie groups of dimension n 4 we have constructed all galilei invariant equations belonging to the class of evolution differential equations under study the list of so obtained invariant equations contains both the well known fourth order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics chemistry and biology

generic hecke algebra for renner monoids abstract to each renner monoid r we associate a generic hecke algebra h r over z q which is a deformation of the monoid z algebra of r if m is a finite reductive monoid with borel subgroup b and associated renner monoid r then we obtain the associated iwahori hecke algebra h m b by specialising q in h r and tensoring by c over z as in the classical case of finite reductive groups

invariants of real forms of affine kac moody lie algebras abstract the theory of vogan diagrams which are dynkin diagrams with an overlay of certain additional information allows one to give a rapid classification of finite dimensional real semisimple lie algebras and to make use of this classification in practice this paper develops a corresponding theory of vogan diagrams for almost compact real forms of indecomposable nontwisted affine kac moody lie algebras in this case also the equivalence classes of vogan diagrams correspond to the isomorphism classes of almost compact real forms although the real forms of such algebras had already been classified the theory of vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification

τ functions and zero curvature equations of toda akns type the connection between functions and zero curvature equations for the homogeneous construction of the basic module l 0 over the simplest affine kac moody algebra a 1 1 is studied

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

irreducible modules for extended affine lie algebras abstract we construct irreducible modules for twisted toroidal lie algebras and extended affine lie algebras this is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules we illustrate our method with examples of extended affine lie algebras of clifford type

q difference equations for askey wilson type integrals via q polynomials abstract in this paper we show how to deduce askey wilson type integrals by the method of q difference equation in addition we construct the relation between bilinear generating functions and solutions of q difference equation more over we generalize bailey s 6 6 summation from the perspective of q integral by the method of q difference equation at last we deduce u n 1 type generating functions for al salam carlitz polynomials by the method of q difference equation