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

julia a fresh approach to numerical computing bridging cultures that have often been distant julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing julia is designed to be easy and fast and questions notions generally held to be laws of nature by practitioners of numerical computing beginlist item high level dynamic programs have to be slow item one must prototype in one language and then rewrite in another language for speed or deployment item there are parts of a system appropriate for the programmer and other parts that are best left untouched as they have been built by the experts endlist we introduce the julia programming language and its design a dance between specialization and abstraction specialization allows for custom treatment multiple dispatch a technique from computer science picks the right algorithm for the right circumstance abstraction which is what good computation is really about recognizes what remains the same after dif

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

identities of the rogers ramanujan bailey type abstract a multiparameter generalization of the bailey pair is defined in such a way as to include as special cases all bailey pairs considered by w n bailey in his paper identities of the rogers ramanujan type proc london math soc 2 50 1949 421 435 this leads to the derivation of a number of elegant new rogers ramanujan type identities

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

an invariant of regular isotopy this paper studies a two variable laurent polynomial invariant of regular isotopy for classical unoriented knots and links this invariant is denoted lk for a link k and it satisfies the axioms 1 regularly isotopic links receive the same polynomial 2 lo0 1 3 l al l a l 4 l small diagrams indicate otherwise identical parts of larger diagrams regular isotopy is the equivalence relation generated by the reidemeister moves of type ii and type iii invariants of ambient isotopy are obtained from l by writhe normalization

lean algebraic multigrid lamg fast graph laplacian linear solver laplacian matrices of graphs arise in large scale computational applications such as semisupervised machine learning spectral clustering of images genetic data and web pages transportation network flows electrical resistor circuits and elliptic partial differential equations discretized on unstructured grids with finite elements a lean algebraic multigrid lamg solver of the symmetric linear system ax b is presented where a is a graph laplacian lamg s run time and storage are empirically demonstrated to scale linearly with the number of edges lamg consists of a setup phase during which a sequence of increasingly coarser laplacian systems is constructed and an iterative solve phase using multigrid cycles general graphs pose algorithmic challenges not encountered in traditional multigrid applications lamg combines a lean piecewise constant interpolation judicious node aggregation based on a new node proximity measure the affinity and an energy correction of coarse level systems this

the bivariate rogers szego polynomials we present an operator approach to deriving mehler s formula and the rogers formula for the bivariate rogers szego polynomials hn x y q the proof of mehler s formula can be considered as a new approach to the nonsymmetric poisson kernel formula for the continuous big q hermite polynomials hn x a q due to askey rahman and suslov mehler s formula for hn x y q involves a 32 sum and the rogers formula involves a 21 sum the proofs of these results are based on parameter augmentation with respect to the q exponential operator and the homogeneous q shift operator in two variables by extending recent results on the rogers szego polynomials hn x q due to hou lascoux and mu we obtain another rogers type formula for hn x y q finally we give a change of base formula for hn x a q which can be used to evaluate some integrals by using the askey wilson integral

numerical methods for gremban s expansion of signed graphs recently a new dimension of graphs has manifested in the network science community that goes beyond the traditional notion of connections and disconnections to include both favorable and adverse relationships there are many applications such as ranking and clustering where data scientists are interested in solving linear systems associated with unsigned graphs we show that there exists a direct relationship between a signed graph and an unsigned variant methods that have been applied for unsigned graphs could be applied to signed graphs in a meaningful way fairly robust solvers for unsigned undirected graph laplacians have been developed but these solvers are not directly applicable to general signed graphs gremban s expansion k gremban combinatorial preconditioners for sparse symmetric diagonally dominant linear systems ph d thesis and tech report cmu cs 96 123 school of computer science carnegie mellon university pittsburgh 1996 is used to transform the signed undirected graph la

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

on lifting q difference operators in the askey scheme of basic hypergeometric polynomials we construct a q difference operator that lifts the continuous q hermite polynomials hn x q of rogers up to the continuous big q hermite polynomials hn x a q on the next level in the askey scheme of basic hypergeometric polynomials this operator is defined as exton s q exponential function eq aqdq in terms of the askey wilson divided q difference operator dq and it represents a particular q extension of the standard shift operator we next show that one can move two steps more upwards in order first to reach the al salam chihara family of polynomials qn x a b q and then the continuous dual q hahn polynomials pn x a b c q in both these cases lifting operators respectively turn out to be convolution type products of two and three one parameter q difference operators of the same type eq aqdq at the initial step at each step we also determine q difference operators that lift the weight function for the continuous q hermite polynomials hn x q successively up to the weight functions for hn x a q qn x a b q and pn x a b c q

positivity preserving transformations for q binomial coefficients several new transformations for q binomial coefficients are found which have the special feature that the kernel is a polynomial with nonnegative coefficients by studying the group like properties of these positivity preserving transformations as well as their connection with the bailey lemma many new summation and transformation formulas for basic hypergeometric series are found the new q binomial transformations are also applied to obtain multisum rogers ramanujan identities to find new representations for the rogers szego polynomials and to make some progress on bressoud s generalized borwein conjecture for the original borwein conjecture we formulate a refinement based on new triple sum representations of the borwein polynomials

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

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

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

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 associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

multilevel adaptive aggregation for markov chains with application to web ranking a multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described the method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggregation disaggregation methods for markov chains in contrast to most existing approaches our aggregation process does not employ any explicit advance knowledge of the topology of the markov chain instead adaptive agglomeration is proposed that is based on the strength of connection in a scaled problem matrix in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level the strength of connection is determined as in the algebraic multigrid method and the aggregation process is fully adaptive with optimized aggregates chosen in each step of the iteration and at all recursive levels the multilevel method is applied to a set of stochastic matrices that provide models for web page ranking numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods the tests also provide more insight into why google s pagerank model is a successful model for determining a ranking of web pages

smoothed aggregation multigrid for markov chains a smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse markov chain it is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for markov chains that have been proposed in the literature the proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems supplemented with a new lumping technique that assures well posedness of the coarse level problems the coarse level operators are singular m matrices on all levels resulting in strictly positive coarse level corrections on all levels numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems both when geometric and algebraic aggregation strategies are used

expansions of one density via polynomials orthogonal with respect to the other abstract we expand the chebyshev polynomials and some of its linear combination in linear combinations of the q hermite the rogers q utraspherical and the al salam chihara polynomials and vice versa we use these expansions to obtain expansions of some densities including q normal and some related to it in infinite series constructed of the products of the other density times polynomials orthogonal to it allowing deeper analysis and discovering new properties on the way we find an easy proof of expansion of the poisson mehler kernel as well as its reciprocal we also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets

tribasic integrals and identities of rogers ramanujan type some integrals involving three bases are evaluated as infinite products using complex analysis many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products the resulting identities are identities of rogers ramanujan type some integer partition interpretations of these identities are given generalizations of the rogers ramanujan type identities involving polynomials are given again as corollaries of integral evaluations

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

subduction coefficients of birman wenzl algebras and racah coefficients of the quantum groups oq n and spq 2m i subduction coefficients irreducible representations of birman wenzl algebras cf r q in the non standard basis are discussed a procedure for evaluating subduction coefficients sdcs or the transformation coefficients between standard and non standard basis of birman wenzl algebras is formulated based on the linear equation method sdcs of cf r q with f 4 are derived racah coefficients of the quantum groups oq n and spq 2m can be obtained from sdcs of birman wenzl algebras cf r q by using the schur weyl brauer duality relation between birman wenzl algebras cf r q and the quantum groups oq n and spq 2m

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

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