modular perverse sheaves on flag varieties ii koszul duality and formality building on the theory of parity sheaves due to juteau mautner williamson we develop a formalism of mixed modular perverse sheaves for varieties equipped with a stratification by affine spaces we then give two applications 1 a koszul type derived equivalence relating a given flag variety to the langlands dual flag variety and 2 a formality theorem for the modular derived category of a flag variety extending a previous result of riche soergel williamson

some koszul rings from geometry abstract we give examples of koszul rings that arise naturally in algebraic geometry in the first part we prove a general result on koszul property associated to an ample line bundle on a projective variety specifically we show how koszul property of multiples of a base point free ample line bundle depends on its castelnuovo mumford regularity in the second part we give examples of koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type

categorifications and cyclotomic rational double affine hecke algebras varagnolo and vasserot conjectured an equivalence between the category mathcal o for crdaha s and a subcategory of an affine parabolic category o of type a we prove this conjecture as applications we prove a conjecture of rouquier on the dimension of simple modules of crdaha s and a conjecture of chuang miyachi on the koszul duality for the category mathcal o of crdaha s

new variable separation solutions of two dimensional burgers system abstract in this paper with the help of g g expansion method we obtain some exact solutions of the riccati equation based on the riccati equation and its exact solutions we find some variable separation solutions with two arbitrary functions of two dimensional burgers system as some special examples these exact solutions can be reduced to variable separation solutions in kink solution soliton solution periodic solution and rational function solution forms

application of the exp function method for nonlinear differential difference equations in this paper we established abundant travelling wave solutions for some nonlinear differential difference equations it is shown that the exp function method with the help of symbolic computation provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics

construction of exact solutions to eigenvalue problems by the asymptotic iteration method we apply the asymptotic iteration method aim ciftci hall and saad 2003 j phys a math gen 36 11807 to solve new classes of second order homogeneous linear differential equation in particular solutions are found for a general class of eigenvalue problems which includes schrodinger problems with coulomb harmonic oscillator or poschl teller potentials as well as the special eigenproblems studied recently by bender et al 2001 j phys a math gen 34 9835 and generalized in the present paper to arbitrary dimension

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

parabolic projective functors in type a abstract we classify projective functors on the regular block of rocha caridi s parabolic version of the bgg category o in type a in fact we show that in type a the restriction of an indecomposable projective functor from o to the parabolic category is either indecomposable or zero as a consequence we obtain that projective functors on the parabolic category o in type a are completely determined up to isomorphism by the linear transformations they induce on the level of the grothendieck group which was conjectured by stroppel in 39

higher symplectic reflection algebras and non homogeneous n koszul property abstract we study a class of associative algebras associated to finite groups acting on a vector space these algebras are non homogeneous n koszul algebra generalizations of symplectic reflection algebras p etingof v ginzburg symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism invent math 147 2002 243 348 the koszul property was generalized to homogeneous algebras of degree n 2 in r berger koszulity for nonquadratic algebras j algebra 239 2001 705 734 in the present paper the extension of the koszul property to non homogeneous algebras is realized through a pbw theorem this pbw theorem is the generalization to the n case of a quadratic result obtained by braverman and gaitsgory a braverman d gaitsgory poincare birkhoff witt theorem for quadratic algebras of koszul type j algebra 181 1996 315 328 see also a polishchuk l positselski quadratic algebras univ lecture ser vol 37 amer math soc providence ri 2005 we work in the general setting where the ground ring is an arbitrary von neumann regular ring

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

tilting modules in category o and sheaves on moment graphs abstract we describe tilting modules of the deformed category o over a semisimple lie algebra as certain sheaves on a moment graph associated to the corresponding block of o we prove that they map to braden macpherson sheaves constructed along the reversed bruhat order under fiebig s localization functor by this means we get character formulas for tilting modules and explain how soergel s result about the andersen filtration gives a koszul dual proof of the semisimplicity of subquotients of the jantzen filtration

whittaker coinvariants for gl m n abstract let w m n be the finite w algebra attached to the principal nilpotent orbit in the general linear lie superalgebra gl m n c in this paper we study the whittaker coinvariants functor which is an exact functor from category o for gl m n c to a certain category of finite dimensional modules over w m n we show that this functor has properties similar to soergel s functor v in the setting of category o for a semisimple lie algebra we also use it to compute the center of w m n explicitly and deduce consequences for the classification of blocks of o up to morita derived equivalence

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

graded decomposition numbers for the blob algebra abstract we give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity

lower central series and free resolutions of hyperplane arrangements if m is the complement of a hyperplane arrangement and a h m k is the cohomology ring of m over a field of characteristic 0 then the ranks k of the lower central series quotients of 1 m can be computed from the betti numbers bii dimtor a k k i of the linear strand in a minimal free resolution of k over a we use the cartan eilenberg change of rings spectral sequence to relate these numbers to the graded betti numbers b dimtor e a k j of a minimal resolution of a over the exterior algebra e from this analysis we recover a formula of falk for 3 and obtain a new formula for 4 the exact sequence of low degree terms in the spectral sequence allows us to answer a question of falk on graphic arrangements and also shows that for these arrangements the algebra a is koszul if and only if the arrangement is supersolvable we also give combinatorial lower bounds on the betti numbers b 1 of the linear strand of the free resolution of a over e if the lower bound is attained for i 2 then it is attained for all i 2 for such arrangements we compute the entire linear strand of the resolution and we prove that all components of the first resonance variety of a are local for graphic arrangements which do not attain the lower bound unless they have no braid sub arrangements we show that b 1 is determined by the number of triangles and k4 subgraphs in the graph

affine jacquet functors and harish chandra categories abstract we define an affine jacquet functor and use it to describe the structure of induced affine harish chandra modules at noncritical levels extending the theorem of kac and kazhdan on the structure of verma modules in the bernstein gelfand gelfand categories o for kac moody algebras this is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine harish chandra modules of lian and zuckerman the latter provides an extension of the works of rocha caridi and wallach a rocha caridi n r wallach projective modules over infinite dimensional graded lie algebras math z 180 1982 151 177 and deodhar gabber and kac v deodhar o gabber v kac structure of some categories of representations of infinite dimensional lie algebras adv math 45 1982 92 116 on block decompositions of bgg categories for kac moody algebras we also derive a compatibility relation between the affine jacquet functor and the kazhdan lusztig tensor product and apply it to prove that the affine harish chandra category is stable under fusion tensoring with the kazhdan lusztig category this compatibility will be further applied in studying translation functors for the affine harish chandra category based on the fusion tensor product

highest weight sl_2 categorifications ii structure theory this paper continues the study of highest weight categorical sl2 actions initi ated in part i we start by refining the definition given there and showing that all examples considered in part i are also highest weight categorifications in the refined sense then we prove that any highest weight sl2 categorification can be filtered in such a way that the suc cessive quotients are so called basic highest weight sl2 categorifications for a basic highest weight categorification we determine minimal projective resolutions of standard objects we use this in particular to examine the structure of tilting objects in basic categorifications and to show that the ringel duality is given by the rickard complex we apply some of these structural results to categories o for cyclotomic rational cherednik algebras we finish by discussing open problems

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

the classification of blocks in bgg category mathcal o o we classify all equivalences between the indecomposable abelian categories which appear as blocks in bgg category mathcal o for reductive lie algebras our classification implies that a block in category mathcal o only depends on the bruhat order of the relevant parabolic quotient of the weyl group as part of the proof we observe that any finite dimensional algebra with simple preserving duality admits at most one quasi hereditary structure

on trivial extensions and higher preprojective algebras abstract for a kosuzl n properly graded algebra we prove that the n 1 preprojective algebra of its quadratic dual is isomorphic to the quadratic dual of a twisted trivial extension this is applied to the slice algebras of stable n translation algebras and gives a noncommutative version of bernstein gelfand gelfand correspondence for such algebras

modular perverse sheaves on flag varieties iii positivity conditions we further develop the general theory of the mixed modular derived category introduced by the authors in a previous paper in this series we then use it to study positivity and q koszulity phenomena on flag varieties

exact three wave solutions for the 3 1 dimensional jimbo miwa equation exact three wave solutions including periodic cross kink wave solutions doubly periodic solitary wave solutions and breather type of two solitary wave solutions for the 3 1 dimensional jimbo miwa equation are obtained using the generalized three wave method it is shown that the generalized three wave method with the help of symbolic computation provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics

a poincare birkhoff witt theorem for quadratic algebras with group actions braverman gaitsgory polishchuk and positselski gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the poincare birkhoff witt property when its homogeneous version is koszul we widen their viewpoint and consider a quotient of an algebra that is free over some not necessarily semisimple subalgebra we show that their theorem holds under a weaker hypothesis we require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra semidirect product algebra of a finite group acting on a koszul algebra obtaining conditions for the poincare birkhoff witt property over nonsemisimple group algebras we prove our main results by exploiting a double complex adapted from guccione guccione and valqui formed from a koszul complex and a resolution of the group giving a practical way to analyze hochschild cohomology and deformations of skew group algebras in positive characteristic we apply these conditions to graded hecke algebras and drinfeld orbifold algebras including rational cherednik algebras and symplectic reflection algebras in arbitrary characteristic with special interest in the case when the characteristic of the underlying field divides the order of the acting group

a generalized koszul theory and its relation to the classical theory abstract let a i 0 a i be a graded locally finite k algebra where a 0 is a finite dimensional algebra whose finitistic dimension is 0 in this paper we develop a generalized koszul theory preserving many classical results and show an explicit correspondence between this generalized theory and the classical theory applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described

koszul duality and equivalences of categories let a and a be dual koszul algebras by positselski a filtered algebra u with gr u a is koszul dual to a differential graded algebra a d we relate the module categories of this dual pair by a hom adjunction this descends to give an equivalence of suitable quotient categories and generalizes work of beilinson ginzburg and soergel

multiple front solutions for the burgers equation and the coupled burgers equations abstract in this work multiple front solutions for the burgers equation and the coupled burgers equations are examined the tanh coth method and the cole hopf transformation are used the work highlights the power of the proposed schemes and the structures of the obtained multiple front solutions

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

some homological properties of category o iii we prove that thick category o associated to a semi simple complex finite dimensional lie algebra is extension full in the category of all modules we also prove the weak alexandru conjecture both for regular blocks of thick category o and the associated categories of harish chandra bimodules but disprove it for singular blocks

unified treatment of the coulomb and harmonic oscillator potentials in d dimensions quantum mechanical models and practical calculations often rely on some exactly solvable models like the coulomb and the harmonic oscillator potentials the d dimensional generalized coulomb potential contains these potentials as limiting cases thus it establishes a continuous link between the coulomb and harmonic oscillator potentials in various dimensions we present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations we define a hilbert space basis the generalized coulomb sturmian basis and calculate the green s operator on this basis and also present an su 1 1 algebra associated with it we formulate the problem for the one dimensional case too and point out that the complications arising due to the singularity of the one dimensional coulomb problem can be avoided with the use of the generalized coulomb potential

extended trial equation method to generalized nonlinear partial differential equations in this article we give the extended trial equation method for solving nonlinear partial differential equations with higher order nonlinearity by use of this method the exact traveling wave solutions including soliton solution singular soliton solutions rational function solution and elliptic integral function solution to one dimensional general improved kdv gikdv equation and r m n equation are obtained also a more general trial equation method is proposed

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

mixed tensors of the general linear supergroup abstract we describe the image of the canonical tensor functor from deligne s interpolating category re p g l m n to r e p g l m n attached to the standard representation this implies explicit tensor product decompositions between any two projective modules and any two kostant modules of g l m n covering the decomposition between any two irreducible g l m 1 representations we also obtain character and dimension formulas for m n we classify the mixed tensors with non vanishing superdimension for m n we characterize the maximally atypical mixed tensors and show some applications regarding tensor products

primary ideals associated to the linear strands of lascoux s resolution and syzygies of the corresponding irreducible representations of the lie superalgebra gl m n abstract we investigate equivariant koszul duality between primary ideals i a b of s s m m n associated to rectangular young diagrams a b and the corresponding atypical irreducible mixed supertensor representations x a b of gl m n in characteristic zero we show i a b to be h 0 s x a b of the koszul dual s x a b and we compute all the higher cohomology of s x a b to be direct sums of i a r b r with multiplicities being given by coefficients of gauss polynomials utilizing this we are able to describe the equivariant syzygies of x a b over s m m n and determine the homological dimension of i a b over s

two block springer fibers of types c and d a diagrammatic approach to springer theory we explain an elementary topological construction of the springer representation on the homology of topological springer fibers of types c and d in the case of nilpotent endomorphisms with two jordan blocks the weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types b and d we determine the decomposition of the representations into irreducibles and relate our construction to classical springer theory as an application we obtain presentations of the cohomology rings of all two block springer fibers of types c and d moreover we deduce explicit isomorphisms between the kazhdan lusztig cell modules attached to the induced trivial module and the irreducible specht modules in types c and d

non relativistic treatment of diatomic molecules interacting with a generalized kratzer potential in hyperspherical coordinates we investigate solutions of a non relativistic wave equation in hyperspherical coordinates for a diatomic molecule system interacting with a generalized kratzer potential rovibrational eigenvalues and corresponding wavefunctions of non relativistic diatomic molecules have been determined within the framework of the asymptotic iteration method certain fundamental conditions for the applications of the asymptotic iteration method such as a suitable asymptotic form for the wave function and the termination condition for the iteration process are discussed n dimensional bound state eigenfunction solutions used in studying the dynamical variables of diatomic molecules are obtained in terms of a confluent hypergeometric function and a generalized laguerre polynomial this systematic approach is tested by calculating the rovibrational energy spectra of hydrogen and sodium chloride molecules

analytical solutions of a generalized non central potential in n dimensions we present that n dimensional non relativistic wave equation for the generalized non central potential with arbitrary angular momentum is analytically solvable in the hyperspherical coordinates asymptotic iteration method as a different approach is applied to obtain n dimensional energy eigenvalues and the corresponding eigenfunctions in hyperspherical coordinates the wave function solutions are obtained in terms of hypergeometric functions and jacobi polynomials the bound states of quantum systems under consideration for some special cases such as hartmann and makarov potentials have been discussed in n dimensions

koszul differential graded algebras and bgg correspondence the concept of koszul differential graded dg for short algebra is introduced in 8 let a be a koszul dg algebra if the ext algebra of a is finite dimensional i e the trivial module a k is a compact object in the derived category of dg a modules then it is shown in 8 that a has many nice properties however if the ext algebra is infinite dimensional little is known about a as shown in 15 see also proposition 2 2 a k is not compact if h a is finite dimensional in this paper it is proved that the koszul duality theorem also holds when h a is finite dimensional by using foxby duality a dg version of the bgg correspondence is deduced from the koszul duality theorem

group actions on algebras and the graded lie structure of hochschild cohomology hochschild cohomology governs deformations of algebras and its graded lie structure plays a vital role we study this structure for the hochschild cohomology of the skew group algebra formed by a nite group acting on an al gebra by automorphisms we examine the gerstenhaber bracket with a view toward deformations and developing bracket formulas we then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory deformations in this context include graded hecke alge bras and symplectic reection algebras we give some general results describing when brackets are zero for polynomial skew group algebras which allow us in particular to nd noncommutative poisson structures for abelian groups we express the bracket using inner products of group characters lastly we inter pret results for graded hecke algebras

skew group algebras of calabi yau algebras abstract the calabi yau property of skew group algebras is discussed it is shown that the skew group algebra a g of a koszul calabi yau algebra a with a finite subgroup g of automorphisms of a is calabi yau if and only if g is a finite subgroup of the special linear group sl a which is defined by means of the homological determinant using the a algebra structure on the yoneda algebra some results in bocklandt et al 2010 bsw are generalized say every connected graded p koszul calabi yau algebra is derived from a superpotential the superpotential for the skew group algebra a g is also constructed

exp function method for solving nonlinear evolution equations with higher order nonlinearity abstract in this paper the exp function method is used to obtain generalized solitary solutions of the generalized drinfel d sokolov wilson dsw system and the generalized 2 1 dimensional burgers type equation then some of the solitary solutions are converted to periodic solutions or hyperbolic function solutions by a simple transformation the results show that the exp function method is a powerful and convenient mathematical tool for solving nonlinear evolution equations with higher order nonlinearity

new variable separation solutions of the 2 1 dimensional asymmetric nizhnik novikov veselov system abstract by means of the exp function method we obtain some new exact solutions of the riccati equation based on the riccati equation and its new exact solutions we find new and more general variable separation solutions with two arbitrary functions of the 2 1 dimensional asymmetric nizhnik novikov veselov annv system as some special examples these new variable separation solutions can degenerate into some known solutions reported in open literature

generating the fukaya categories of hamiltonian g manifolds abstract let g be a compact lie group and let k be a field of characteristic p geq 0 such that h g has no p torsion if p 0 we show that a free lagrangian orbit of a hamiltonian g action on a compact monotone symplectic manifold x split generates an idempotent summand of the monotone fukaya category mathcal f x k if and only if it represents a nonzero object of that summand slightly more general results are also provided our result is based on an explicit understanding of the wrapped fukaya category mathcal w t g k through koszul twisted complexes involving the zero section and a cotangent fibre and on a functor d b mathcal w t g k to d b mathcal f x times x k canonically associated to the hamiltonian g action on x we explore several examples which can be studied in a uniform manner including toric fano varieties and certain grassmannians

exact solutions of the drinfel d sokolov wilson equation using the exp function method abstract the generalized solitary solutions of the classical drinfel d sokolov wilson equation dswe are obtained using the exp function method then some of these solutions are easily converted into kink shaped solutions and blow up solutions by a simple transformation

categories of modules over an affine kac moody algebra and finiteness of the kazhdan lusztig tensor product abstract to each category c of modules of finite length over a complex simple lie algebra g closed under tensoring with finite dimensional modules we associate and study a category aff c of smooth modules in the sense of kazhdan and lusztig d kazhdan g lusztig tensor structures arising from affine lie algebras i j amer math soc 6 1993 905 947 of finite length over the corresponding affine kac moody algebra in the case of central charge less than the critical level equivalent characterizations of these categories are obtained in the spirit of the works of kazhdan and lusztig d kazhdan g lusztig tensor structures arising from affine lie algebras i j amer math soc 6 1993 905 947 and lian and zuckerman b h lian g j zuckerman brst cohomology and noncompact coset models in proceedings of the xxth international conference on differential geometric methods in theoretical physics new york 1991 1992 pp 849 865 b h lian g j zuckerman an application of infinite dimensional lie theory to semisimple lie groups in representation theory of groups and algebras in contemp math vol 145 1993 pp 249 257 in the main part of this paper we establish a finiteness result for the kazhdan lusztig tensor product which can be considered as an affine version of a theorem of kostant b kostant on the tensor product of a finite and an infinite dimensional representation j funct anal 20 1975 257 285 it contains as special cases the finiteness results of kazhdan lusztig d kazhdan g lusztig tensor structures arising from affine lie algebras i j amer math soc 6 1993 905 947 and finkelberg m finkelberg phd thesis harvard university 1993 and states that for any subalgebra f of g which is reductive in g the affinization of the category of finite length admissible g f modules is stable under kazhdan lusztig s tensoring with the affinization of the category of finite dimensional g modules which is o in the notation of d kazhdan g lusztig tensor structures arising from affine lie algebras i j amer math soc 6 1993 905 947 d kazhdan g lusztig tensor structures arising from affine lie algebras ii j amer math soc 6 1994 949 1011 d kazhdan g lusztig tensor structures arising from affine lie algebras iv j amer math soc 7 1994 383 453

the g g expansion method and travelling wave solutions for a higher order nonlinear schrodinger equation abstract the higher order nonlinear schrodinger equation is reduced to a nonlinear ordinary differential equation ode by using a simple transformation various solutions of the nonlinear ode are obtained by the g g expansion method proposed recently with the aid of solutions of the nonlinear ode more explicit travelling wave solutions involving two arbitrary parameters of the higher order nonlinear schrodinger equation are found out the travelling waves are expressed by the hyperbolic functions trigonometric functions and rational functions when the parameters are taken as special values the solitary waves are also derived from the travelling waves

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

cyclotomic double affine hecke algebras and affine parabolic category o using the orbifold kz connection we construct a functor from an affine parabolic category o of type a to the category o of a cyclotomic rational double affine hecke algebra h then we give several results concerning this functor

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

recurrence relations for radial wavefunctions for the nth dimensional oscillators and hydrogenlike atoms we present a general procedure for finding recurrence relations of the radial wavefunctions for nth dimensional isotropic harmonic oscillators and hydrogenlike atoms

soliton solutions of the korteweg de vries and the kadomtsev petviashvili equations the wronskian technique the well known soliton solutions of the kadomtsev petviashvili equations are written in terms of determinants of wronskian form by using this compact representation together with the hirota bilinear form of the equations it is demonstrated by elementary algebraic methods that the n soliton solution satisfies the evolution equation and the n and n 1 soliton solutions satisfy the associated backlund transformation the relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the n soliton solution is also given

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

asymptotic iteration method for eigenvalue problems an asymptotic iteration method for solving second order homogeneous linear differential equations of the form y 0 x y s0 x y is introduced where 0 x 0 and s0 x are c functions applications to schrodinger type problems including some with highly singular potentials are presented