the q deformed boson realization of representations of quantum universal enveloping algebras for q a root of unity i the case of uqsl l the properties of q deformed boson operators with non generic q q is a root of unity are analysed by using the representation theory method and their finite dimensional representations are thereby obtained based on this discussion reducibilities and decompositions of q deformed boson realized representations of quantum universal enveloping algebra uqsl l are studied for non generic cases the explicit matrix elements of some indecomposable representations are obtained on the q deformed fock spaces necessary details are provided for uqsl 2 and uqsl 3 in particular the lusztig operator extension of uqsl 2 is discussed in an explicit form

approximation schemes for infinite linear programs this paper presents approximation schemes for an infinite linear program in particular it is shown that under suitable assumptions the program s optimum value can be approximated by the values of finite dimensional linear programs and that in addition every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution for the original problem

some quantum lie algebras of type dn positive a quantum lie algebra is constructed within the positive part of the drinfeld jimbo quantum group of type dn our quantum lie algebra structure includes a generalized antisymmetry property and a generalized jacobi identity closely related to the braid equation a generalized universal enveloping algebra of our quantum lie algebra of type dn positive is proved to be the drinfeld jimbo quantum group of the same type the existence of such a generalized lie algebra is reduced to an integer programming problem moreover when the integer programming problem is feasible we show by means of the generalized jacobi identity that the poincare birkhoff witt theorem basis is still true

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

a q deformation of wakimoto modules primary fields and screening operators theq vertex operators of frenkel and reshetikhin are studied by means of aq deformation of the wakimoto module for the quantum affine algebrau q widehat mathfrak s mathfrak l 2 at an arbitrary levelk 0 2 a fock module version of theq deformed primary field of spinj is introduced as well as the screening operators which anti commute with the action ofu q widehat mathfrak s mathfrak l 2 up to a total difference of a field a proof of the intertwining property is given for theq vertex operators corresponding to the primary fields of spinj 1 2z 0 a sample calculation of the correlation function is also given

symmetries of clebsch gordan coefficients of the quantum group uq n permutation symmetries for clebsch gordan coefficients of the quantum group uq n and for uq n 1 reduced clebsch gordan coefficients of uq n are derived in particular the formula relating clebsch gordan coefficients for the tensor product tm1 tm2 with clebsch gordan coefficients for the tensor product with permuted representations tm1 and tm is obtained where tm is one of the irreducible representations in the decomposition of tm1 tm2 contrary to the classical case coefficients connecting these clebsch gordan coefficients depend on gel fand tsetlin patterns of one of irreducible representations this dependence is explicitly determined

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

q deformation of the lorentz group we describe a q deformation of the lorentz group in terms of a q deformation of the van der waerden spinor algebra

a quantum analog of the z algebra we define a natural quantum analog for the z algebra and which we refer to as the zq algebra by modding out the heisenberg algebra from the quantum affine algebra uq sl 2 with level k we discuss the representation theory of this zq algebra in particular we exhibit its reduction to a group algebra and to a tensor product of a group algebra with a quantum clifford algebra when k 1 and k 2 and thus we recover the explicit constructions of uq sl 2 standard modules as achieved by frenkel jing and bernard respectively moreover for arbitrary nonzero level k we show that the explicit basis for the simplest z generalized verma module as constructed by lepowsky and primc is also a basis for its corresponding zq module i e it is invariant under the q deformation for generic q we expect this zq algebra associated with uq sl 2 at level k to play the role of a dynamical symmetry in the off critical zk statistical models

toroidal and level 0 uq sln 1 actions on uq gln 1 modules 1 utilizing a braid group action on a completion of uq sln 1 an algebra homomorphism from the toroidal algebra uq sln 1 tor n 2 to a completion of uq gln 1 is obtained 2 the toroidal actions by saito induces a level 0 uq sln 1 action on level 1 integrable highest weight modules of uq sln 1 another level 0 uq sln 1 action was defined by jimbo et al in the case n 1 using the fact that the intertwiners of uq sln 1 modules are intertwiners of toroidal modules for an appropriate comultiplication the relation between these two level 0 uq sln 1 actions is clarified

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

the cyclic representations of the quantum algebra uq osp 2 1 in terms of the zn algebra the quantum superalgebra uq osp 2 1 is embedded into the zn algebra and its cyclic representations are presented in the nongeneric case

quantized enveloping algebras for borcherds superalgebras the rst author was supported in part by nsf grant dms 9300523 the second author was supported in part by the nondirected research fund korea research foundation 1996 the third author was supported in part by a faculty research grant from st lawrence university

the q deformed boson realization of representation of quantum universal enveloping algebras for q a root of unity ii the subalgebra chain of uq cl for pt i see ibid vol 24 p 3265 1991 applying the method of the q deformed boson realization for the quantum group to the case where q is a root of unity the authors establish a set of standard basis for the representation space of the quantum universal enveloping algebra uq cl associated with a typical subalgebra chain of uq cl on this basis they systematically obtain irreducible and indecomposable representations of uq cl and its subalgebras they discuss uq c2 in detail

non integrable representations of the restricted quantum analogue of sl 3 the structure of irreducible representations of restricted at roots of unity is understood within the gelfand zetlin basis the latter needs a weakened definition for non integrable representations where the quadratic casimir operator of the quantum subalgebra is not completely diagonalized this is necessary in order to take into account the indecomposable modules that appear the set of redefined mixed states has a teepee shape inside the pyramid made with the whole representation

quantum howe duality and invariant polynomials abstract we construct two examples of q deformed classical howe dual pairs sl 2 c so 3 c and sl 2 c sl n c moreover we obtain a noncommutative version of the first fundamental theorem of classical invariant theory our approach to these dualities differs from 9 and 11 furthermore we solve the tensor product decomposition problem for verma modules over u q sl 2 c provided q is not a root of unity

singular vectors of representations of quantum groups the author gives explicit formulae for singular vectors of verma modules over uq g they give the general formula for g an for some cases when g dn and for some rank two subalgebras of g not an dn for this he uses a special basis of uq g where g is the negative root subalgebra of g which was introduced in his earlier work on the case q 1 this basis seems more economical than the poincare birkhoff witt type of basis used by malikov feigin and fuchs 1986 for the construction of singular vectors of verma modules in the case q 1 furthermore this basis turns out to be part of a general basis introduced recently for other reasons by lusztig 1988 for uq b where b is a borel subalgebra of g

optimal solution of a monge kantorovitch transportation problem abstract suppose that p and q are probabilities on a separable banach space e it is known that if p q satisfies certain regularity conditions and a random variable x has law p then there exists a function f e e such that the function f x has the law q and the random pair x f x is an optimal coupling for the monge kantorovitch problem in this paper we provide an approximation of the function f when the law q is discrete thenwe extend this main result to any law q the proofs are based on a relationship between optimal couplings and nonlinear equations

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

classical invariant theory for the quantum symplectic group in this paper we establish the quantum analogues of the first and second fundamental theorem for vector invariants for the symplectic group let us briefly recall this result consider a vector space v endowed with a non degenerate antisymmetric bilinear form let sp v denote the group of isometries the first and second fundamental theorems for vector invariants for sp v tells us that any invariant polynomial function on

covariant realization of quantum spaces as star products by drinfeld twists covariance of a quantum space with respect to a quantum enveloping algebra ties the deformation of the multiplication of the space algebra to the deformation of the coproduct of the enveloping algebra since the deformation of the coproduct is governed by a drinfeld twist the same twist naturally defines a covariant star product on the commutative space however this product is in general not associative and does not yield the quantum space it is shown that there are certain drinfeld twists which realize the associative product of the quantum plane quantum euclidean four space and quantum minkowski space these twists are unique up to a central two coboundary the appropriate formal deformation of real structures of the quantum spaces is also expressed by these twists

quantum line bundles on s2 and method of orbits for suq 2 the quantum sphere is covered by two quantum complex planes quantum line bundle on s2 is introduced by specifying a transition function the structure of the quantum complex planes enables one to construct representations of suq 2 acting in spaces of holomorphic sections in the quantum case dressing transformation replaces coadjoint action

exact solutions to the transportation problem on the line given distributions w of production and v of consumption on the line the monge kantorovich problem is to decide which producer should supply each consumer in order to minimize the total transportation costs here cost will be assumed to be a strictly concave function of the distance which translates into an economy of scale for longer trips and may encourage cross hauling the resulting solutions display a hierarchical structure that reflects a striking separation into local and global scales also found in the real world moreover this structure can be exploited to reduce the infinite dimensional linear problem to a convex minimization in m variables where 2 m 2 counts the number of times that w v changes sign a combinatorial algorithm is then derived which yields exact solutions by optimizing a certain finite sequence of convex separable network flows

classical flag varieties for quantum groups the standard quantum sl n c abstract we suggest a possible programme to associate geometric flag like data to an arbitrary simple quantum group in the spirit of the noncommutative al gebraic geometry developed by artin tate and van den bergh we then carry out this programme for the standard quantum sl n of drinfel d and jimbo where the varieties involved are certain t stable subvarieties of the ordinary flag variety

a remark on the representation theory of the algebra uq sl n when q is a root of unity an associative algebra is defined on the set of certain irreducible representation spaces of the algebra uq sl n when q is a root of unity this associative algebra is shown to be connected with some algebras defined by the decomposition rule of tensor products of the irreducible representations of certain finite groups

quantumr matrices related to the spin representations ofbn anddn we present the explicit form of the trigonometricr matrices related to the spin representations of the simple lie algebrasx n b n d n we conjecture that one dimensional configuration sums of the corresponding vertex models in the face formulation are the string functions ofx 1 modules

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

the spin 1 2xxz heisenberg chain the quantum algebra uq sl 2 and duality transformations for minimal models the finite size scaling spectra of the spin 1 2 xxz heisenberg chain with toroidal boundary conditions and an even number of sites provide a projection mechanism yielding the spectra of models with a central charge c 1 including the unitary and nonunitary minimal series taking into account the half integer angular momentum sectors which correspond to chains with an odd number of sites in many cases leads to new spinor operators appearing in the projected systems these new sectors in the xxz chain correspond to new types of frustration lines in the projected minimal models the corresponding new boundary conditions in the hamiltonian limit are investigated for the ising model and the 3 state potts model and are shown to be related to duality transformations which are an additional symmetry at their self dual critical point by different ways of projecting systems we find models with the same central charge sharing the same operator content and modular invariant partition function which however differ in the distribution of operators into sectors and hence in the physical meaning of the operators involved related to the projection mechanism in the continuum there are remarkable symmetry properties of the finite xxz chain the observed degeneracies in the energy and momentummore spectra are shown to be the consequence of intertwining relations involving u sub q sl 2 quantum algebra transformations 35 refs 3 tabs less

the center of the quantized matrix algebra abstract we compute the center in the case where q is a root of unity the main steps are to compute the degree of an associated quasipolynomial algebra and to compute the dimensions of some interesting irreducible modules

ringel hall algebras and lusztig s symmetries abstract let d be the double ringel hall algebra of a finite dimensional hereditary algebra the present paper shows that the bgp reflection operators of d coincide with lusztig s symmetries up to canonical isomorphisms and satisfy the braid group relations on the whole double ringel hall algebra d this answers a question of sevenhant and van den bergh j algebra 221 1999 135 160

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

raising and lowering operators for uq n after reviewing the defining relations for the gq algebra uq n the author constructs a cartan weyl basis and lists the q commutation relations of its generators he then uses the latter to explicitly construct sets of raising and lowering operators for the canonical chain of q algebras uq n uq n 1 generalizing those introduced by nagel and moshinsky 1965 for u m u n 1 finally he gives their normalization coefficients and shows how the normalized operators can be used to go from any gel fand tseitlin basis state that is of highest weight in uq n 1 to any other one and ultimately to construct the whole gel fand tseitlin basis from its highest weight state this work both generalizes a previous work of ueno et al 1989 on lowering operators and provides explicit expressions for their operators in terms of uq n cartan weyl generators

finite dimensional representations of quantum affine algebras we give a practical method for the construction of finite dimensional representations of uq g where g is a non twisted affine kac moody algebra with no derivation and zero central charge it is well known that the finite dimensional representations of a quantum group uq g at generic q are simply deformations of the corresponding irreps of the classical lie algebra g this is however no longer true when one goes over to an affine kac moody algebra g and its q deformation uq g in most cases it is necessary to take the direct sum of several irreducible uq g modules to form an irreducible uq g module which then becomes reducible at q 1 we illustrate our technique by working out explicit examples for g c2 and g g2 these finite dimensional modules may for example determine the multiplet structure of quantum solitons in affine toda theory

a general duality theorem for marginal problems given probability spaces xi ai pi i 1 2 letm p1 p2 denote the set of all probabilities on the product space with marginalsp1 andp2 and leth be a measurable function on x1 x2 a1 a2 in order to determine supfh dp where the supremum is taken overp inm p1 p2 a general duality theorem is proved only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions an example without any further topological or tightness assumptions an example is given to show that the assumption of perfectness is essential applications to probabilities with given marginals and given supports stochastic order and probability metrics are included

quantum group structure in the fock space resolutions of widehat sl n representations we describe a complex of wakimoto type fock space modules for the affine kac moody algebra widehat sl n the intertwining operators that build the complex are obtained from contour integrals of so called screening operators we show that a quantum group structure underlies the algebra of screening operators this observation greatly facilitates the explicit determination of the intertwiners we conjecture that the complex provides a resolution of an irreducible highest weight module in terms of fock spaces

on a class of infinite dimensional representations of uq sl 2 a class of infinite dimensional representations of the quantum universal enveloping algebra uq sl 2 is considered equivalence and reducibility conditions are examined

on solutions to the mass transfer problem this paper studies the monge kantorovich mass transfer mt problem on metric spaces with possibly unbounded cost function conditions are given under which the mt problem is solvable and furthermore an optimal solution can be obtained as the weak limit of a sequence of optimal solutions to suitably approximating mt problems

space time trade offs for finding shortest unique substrings and maximal unique matches abstract given a string x 1 n and a position k 1 n a shortest unique substring of x covering k denoted by s k is a substring x i j of x which satisfies the following conditions i i k j ii i is the only position where there is an occurrence of x i j and iii j i is minimized current best known algorithms for finding s k require n words of working space and o n time let be a given parameter we present the following new results given a k 1 n we can compute s k in o n 2 log n time using x and an additional o n words of working space for every k 1 n we can compute s k in o n 2 log n time using x and an additional o n words and 4 n o n bits of working space we present an o n log c 1 n time randomized algorithm that uses n log c n words in addition to that mentioned above where c 0 is an arbitrary constant in this case the reported string is unique and covers k but with probability at most n o 1 may not be the shortest by choosing 1 our results imply the first sub linear space in addition to the input string solution to these problems we also present the following two results an algorithm that finds s k for every k 1 n using o n log bits of working space in o n log n time where n is the number of distinct symbols in x a 4 n o n bit index that can report s k for any k in o 1 time as a consequence of our techniques we also obtain similar space and time tradeoffs for a related problem of finding maximal unique matches of two strings delcher et al 1999 14

quantum b functions of prehomogeneous vector spaces of commutative parabolic type we show that there exist natural q analogues of the b functions for the prehomogeneous vector spaces of commutative parabolic type and calculate them explicitly in each case our method of calculating the b functions seems to be new even for the original case q 1

faces of highest weight modules and the universal weyl polyhedron abstract let v be a highest weight module over a kac moody algebra g and let conv v denote the convex hull of its weights we determine the combinatorial isomorphism type of conv v i e we completely classify the faces and their inclusions in the special case where g is semisimple this brings closure to a question studied by cellini and marietti 2015 7 for the adjoint representation and by khare 2016 2017 17 18 for most modules the determination of faces of finite dimensional modules up to the weyl group action and some of their inclusions also appears in previous works of satake 1960 25 borel and tits 1965 3 vinberg 1990 26 and casselman 1997 6 for any subset of the simple roots we introduce a remarkable convex cone which we call the universal weyl polyhedron which controls the convex hulls of all modules parabolically induced from the corresponding levi factor namely the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules as well as how faces degenerate as the highest weight gets increasingly singular to our knowledge this cone is new in finite and infinite type we further answer a question of michel brion by showing that the localization of conv v along a face is always the convex hull of the weights of a parabolically induced module finally as we determine the inclusion relations between faces representation theoretically from the set of weights without recourse to convexity we answer a similar question for highest weight modules over symmetrizable quantum groups

representations of the quantum algebra uq su1 1 all irreducible representations of the quantum group uq su1 1 are given they are determined by two complex numbers infinitesimally unitary representations are separated from the set of irreducible representations it is shown that the symmetric operators of irreducible infinitesimally unitary representations corresponding to generators of the lie algebra su1 1 admit self adjoint extensions

a classification of unitary highest weight modules of the quantum analogue of the symmetric pair an an 1 abstract we study the irreducibility and unitarity of highest weight modules of the quantized enveloping algebra associated to s u n 1 we obtain a classification of these modules analogous to that of ehw in the classical context we also compute all commutation relations between root vectors in u q s u n 1

breaking a time and space barrier in constructing full text indices suffix trees and suffix arrays are the most prominent full text indices and their construction algorithms are well studied in the literature the fastest algorithm runs in o n time while it requires o n log n bit working space where n denotes the length of the text on the other hand the most space efficient algorithm requires o n bit working space while it runs in o n log n time it was open whether these indices can be constructed in both o n log n time and o n log n bit working space this paper breaks the above time and space barrier under the unit cost word ram we give an algorithm for constructing the suffix array which takes o n time and o n bit working space for texts with constant size alphabets note that both the time and the space bounds are optimal for constructing the suffix tree our algorithm requires o n log epsilon n time and o n bit working space for any 0 epsilon 1 apart from that our algorithm can also be adopted to build other existing full text indices such as compressed suffix tree compressed suffix arrays and fm index we also study the general case where the size of the alphabet sigma is not constant our algorithm can construct a suffix array and a suffix tree using optimal o n log sigma bit working space while running in o n log log sigma time and o n log epsilon n log sigma time respectively these are the first algorithms that achieve o n log n time with optimal working space moreover for the special case where log sigma o log log n 1 epsilon we can speed up our suffix array construction algorithm to the optimal o n

strong duality of the monge kantorovich mass transfer problem in metric spaces this paper studies the monge kantorovich mass transfer mt problem on metric spaces and with an unbounded cost function conditions are given under which the strong duality condition holds that is mt and its dual mt are both solvable and their optimal values coincide

the fusion of r matrices using the birman wenzl murakami algebra the action of the birman wenzl murakami algebra is used as the centralizer of soq n and spq n to fuse jimbo s trigonometric r matrices and thus to construct new solutions of the yang baxter equation

parallel computation of the burrows wheeler transform in compact space abstract the burrows wheeler transform bwt has become since its introduction a key tool for representing large text collections in compressed space while supporting indexed searching on a text of length n over an alphabet of size it requires o n lg bits of space instead of the o n lg n bits required by classical indexes a challenge for its adoption is to build it within o n lg bits as well there are some sequential algorithms building it within this space but no such a parallel algorithm in this paper we present a pram crew algorithm to build the bwt using o n lg n work o lg 3 n lg depth and o n lg bits

irregular uq sl 3 representations at roots of unity via gel fand weyl zetlin basis the gel fand weyl zetlin gwz description of the uq sl 3 irregular irreps at roots of unity is explicitly given those are irreps fixed by the same parameters as the unitary irreducible representations uirs of su 3 yet having dimensions smaller than their classical counterparts the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule this description is made geometrically transparent by an arrangement of the standard su 3 gwz basis in a hexagonal pyramid which is valid for any q and seems new even for q 1 the pyramid has as a base the standard hexagon which gives the weight space of the uirs of su 3 in the plane of third component of isospin iz and hypercharge y while third dimension of this pyramid is related to the isospin i algebraically this arrangement is related to a one to one correspondence between the abstract gwz states and monomials in the algebra of raising generators uq g however those monomials are not in the standard poi

integrals quantum galois extensions and the affineness criterion for quantum yetter drinfel d modules abstract in this paper we shall generalize the notion of an integral on a hopf algebra introduced by sweedler by defining the more general concept of an integral of a threetuple h a c where h is a hopf algebra coacting on an algebra a and acting on a coalgebra c we prove that there exists a total integral c hom c a of h a c if and only if any representation of h a c is injective in a functorial way as a corepresentation of c in particular the quantum integrals associated to yetter drinfel d modules are defined let now a be an h bicomodule algebra h y d a the category of quantum yetter drinfel d modules and b a a s 1 a 1 a 1 a 0 1h a the subalgebra of coinvariants of the verma structure a h y d a we shall prove the following affineness criterion if there exists h hom h a a total quantum integral and the canonical map a b a h a a b b s 1 b 1 b 1 ab 0 is surjective i e a b is a quantum homogeneous space then the induction functor b a m b h y d a is an equivalence of categories the affineness criteria proven by cline parshall and scott and independently by oberst for affine algebraic groups schemes and schneider in the noncommutative case are recovered as special cases

quantization of lie group and algebra of g2 type in the faddeev reshetikhin takhtajan approach based on the quantized universal enveloping que algebras a quantization of the automorphism group of some nonassociative algebras is given in the formulation employing noncommuting matrix entries a quantum group of g2 type included in this scheme is studied in detail in the faddeev reshetikhin takhtajan frt approach also in the formulation employing noncommuting matrix entries the que algebra of g2 type is reconstructed through the pairing induced by the r matrix between the quantum group and the que algebra