lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

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

formality theorem for lie bialgebras and quantization of twists and coboundary r matrices abstract let g be a lie bialgebra let u g a quantization of g through etingof kazhdan functor we prove the existence of a l morphism between the lie algebra c g g and the tensor algebra without unit t u t u g 1 with lie algebra structure given by the gerstenhaber bracket when s is a twist for g we deduce from the formality morphism the existence of a quantum twist f when g r is a coboundary lie bialgebra we get the existence of a quantization r of r

projective representations of symmetric groups via sergeev duality in this article we determine the irreducible projective representations of the symmetric group sd and the alternating group ad over an algebraically closed field of characteristic p 6 2 these matters are well understood in the case p 0 thanks to the fundamental work of schur 24 in 1911 as well as the much more recent work of nazarov 19 20 sergeev 25 26 and others so the focus here is primarily on the case of positive characteristic where surprisingly little is known as yet in particular we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set rpp d of restricted p strict partitions of d such partitions arose recently in work of kashiwara miwa peterson and yung 11 and leclerc and thibon 14 on the q deformed fock space of the affine kac moody algebra of type a p 1 leclerc and thibon proposed that rpp d should label the irreducible projective representations in some natural way and we show here how this can be done note that for p 3 5 the labelling problem was solved in 1 3 while if p 2 all projective representations of sd and ad are linear so do not need to be considered further here to be more precise recall that is a partition of d if 1 2 is a non increasing sequence of non negative integers summing to d call p strict if in addition

formality for lie algebroids using dolgushev s generalization of fedosov s method for deformation quantization we give a positive answer to a question of p xu can one prove a formality theorem for lie algebroids as a direct application of this result we obtain that any triangular lie bialgebroid is quantizable

representation theory of 0 hecke clifford algebras abstract the representation theory of 0 hecke clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning bergeron hivert and thibon have proved that the grothendieck ring of the category of finitely generated supermodules of 0 hecke clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by stembridge in this paper we further study the category of finitely generated projective supermodules and clarify the correspondence between it and the peak algebra of symmetric groups in particular two kinds of restriction rules for induced projective supermodules are obtained after that we consider the corresponding heisenberg double and its fock representation to prove that the ring of peak quasisymmetric functions is free over its subring of symmetric functions spanned by schur s q functions

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

quantization of formal classical dynamical r matrices the reductive case abstract in this paper we prove the existence of a formal dynamical twist quantization for any triangular and non modified formal classical dynamical r matrix in the reductive case the dynamical twist is constructed as the image of the dynamical r matrix by a l quasi isomorphism this quasi isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence

an exact solution of the schrodinger equation for a multiterm potential the authors show that the schrodinger equation with the potential bx cx2 dx3 ex4 is exactly solvable on the half line x or 0 provided two simple relations between b c d and e hold some remarks concerning the b d 0 case are made

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

căldăraru s conjecture and tsygan s formality in this paper we complete the proof of caldararu s conjecture on the compatibility between the module structures on differential forms over poly vector fields and on hochschild homology over hochschild cohomology in fact we show that twisting with the square root of the todd class gives an isomorphism of precalculi between these pairs of objects our methods use formal geometry to globalize the local formality quasi isomorphisms introduced by kontsevich and shoikhet the existence of the latter was conjectured by tsygan we also rely on the fact recently proved by the first two authors that shoikhet s quasi isomorphism is compatible with cap products after twisting with a maurer cartan element

the recovery of even polynomial potentials we consider the problem of reconstructing an even polynomial potential from one set of spectral data of a sturm liouville problem we show that we can recover an even polynomial of degree 2m from m 1 given taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum the idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function we then compute these coefficients by solving a nonlinear algebraic system and provide numerical examples at the end because of its algebraic nature the method applies also to non self adjoint problems

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

fisher information of special functions and second order differential equations we investigate a basic question of analytic information theory namely the evaluation of the fisher information and the relative fisher information with respect to a non negative function for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second order differential equations we obtain explicit expressions for these information theoretic properties via the expectation values of the coefficients of the differential equation we illustrate our approach for various nonrelativistic d dimensional wavefunctions and some special functions of physicomathematical interest emphasis is made in the nikiforov uvarov hypergeometric type functions which include and generalize the hermite functions and the gauss and kummer hypergeometric functions among others

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

non linear characterizations for functions of hypergeometric type and their derivatives of any order abstract to find characterizations for special functions is an important question in applied mathematics and mathematical physics up to now most effort has been invested for polynomial sets which are orthogonal on the real line in this paper two non linear characterizations for a class of functions the so called functions of hypergeometric type are found not only for the functions themselves but also for their derivatives of any order members of this class are the bessel functions the hypergeometric functions the classical orthogonal polynomials as well as many other functions of mathematics and physics applications to the subclass of polynomials of hypergeometric type are made with emphasis on the classical orthogonal polynomials for which the coefficients of the above mentioned non linear characterizations are tabulated

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

the algebra of quasi symmetric functions is free over the integers let z denote the leibniz hopf algebra which also turns up as the solomon descent algebra and the algebra of noncommutative symmetric functions as an algebra z z z1 z2 the free associative algebra over the integers in countably many indeterminates the coalgebra structure is given by zn ni 0 zi zn i z0 1 let m be the graded dual of z this is the algebra of quasi symmetric functions the ditters conjecture says that this algebra is a free commutative algebra over the integers in this paper the ditters conjecture is proved

algebraic structures on grothendieck groups of a tower of algebras abstract the grothendieck group of the tower of symmetric group algebras has a self dual graded hopf algebra structure inspired by this we introduce by way of axioms a general notion of a tower of algebras and study two grothendieck groups on this tower linked by a natural paring using representation theory we show that our axioms give a structure of graded hopf algebras on each grothendieck groups and these structures are dual to each other we give some examples to indicate why these axioms are necessary we also give auxiliary results that are helpful to verify the axioms we conclude with some remarks on generalized towers of algebras leading to a structure of generalized bialgebras in the sense of loday on their grothendieck groups

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

integrability of hamiltonians with polynomial potentials in this work we study dynamical systems with polynomial potentials such as those of henon heiles yang mills and various generalizations by means of the nonintegrability theory developed by the authors all these problems have also been investigated by using other theories like those proposed by ziglin yoshida morales or the painleve analysis in the examples considered our method allows us to reproduce with quite less work or even to improve the results obtained by other authors

the klein gordon equation with ring shaped potentials asymptotic iteration method this study presents the solutions of three dimensional klein gordon equation for the spherically and non spherically harmonic oscillatory ring shaped potentials within the framework of asymptotic iteration method using the method of variable separation this study obtains the radial and angular equations and then the bound states energy eigenvalues and corresponding eigenfunctions are obtained analytically

a class of exactly solvable rationally extended non central potentials in two and three dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

quantum quasi symmetric functions and hecke algebras the algebra of quasi symmetric functions is known to describe the characters of the hecke algebra of type at v 0 we present a quantization of this algebra defined in terms of filtrations of induced representations of the 0 hecke algebra we show that this q deformed algebra admits a simple realization in terms of quantum polynomials for generic values of q the algebra of quantum quasi symmetric functions is isomorphic to the one of noncommutative symmetric functions this gives rise to a one parameter family of hilbert space structures on the algebra of noncommutative symmetric functions as well as to new interesting bases

the peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

formality theorems for hochschild chains in the lie algebroid setting in this paper we prove lie algebroid versions of tsygan s formality conjecture for hochschild chains both in the smooth and holomorphic settings in the holomorphic setting our result implies a version of tsygan s formality conjecture for hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary poisson lie algebroid the proofs are based on the use of kontsevich s quasi isomorphism for hochschild cochains of r y 1 y d shoikhet s quasi isomorphism for hochschild chains of r y 1 y d and fedosov s resolutions of the natural analogues of hochschild co chain complexes associated with a lie algebroid

a unified treatment of the non relativistic and relativistic hydrogen atom iii the reduced green functions for pt i see ibid p 79 94 this is the second in a series of three papers in which it is shown how the radial part of non relativistic and relativistic hydrogenic bound state calculations involving the green functions can be presented in a unified manner in this paper the nonrelativistic green function is examined in detail new functional forms are presented and a clear mathematical progression is shown to link these and most other known forms a linear transformation of the four radial parts of the relativistic green function is given which allows for the presentation of this function as a simple generalization of the non relativistic green function thus many properties of the non relativistic green function are shown to have simple relativistic generalizations in particular new recursion relations of the radial parts of both the non relativistic and relativistic green functions are presented along with new expressions for the double laplace transforms and recursion relations between the radial matrix elements a novel proof of the sturmian form of the radial green functions is given in an appendix

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

the three term recurrence relation and the differentiation formulas for hypergeometric type functions abstract the functions of hypergeometric type are the solutions y y x of the differential equation z y z y y 0 where are polynomials of degrees not higher than 2 and 1 respectively and is a constant here we consider a class of functions of hypergeometric type with the additional condition that 1 2 1 0 being a complex number in general moreover we assume that the coefficients of the polynomials and have no dependence on to this class of functions belong gauss kummer and hermite functions the classical orthogonal polynomials and many other functions encountered in linear and non linear physics we obtain two important structural properties of these functions i the so called three term recurrence relation which correlates three functions of successive orders and ii the differentiation formulas also called ladder or structure relations or even differential recurrence relations which relate the first derivative y z with the functions y z and y 1 z or y 1 z finally these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y where is a positive integer number and the associated contour is closed for completeness the explicit expressions corresponding to all classical orthogonal polynomials jacobi laguerre hermite and bessel are tabulated

a relativistic model of the n dimensional singular oscillator an exactly solvable n dimensional model of the quantum isotropic singular oscillator in the relativistic configurational space is proposed it is shown that through the simple substitutions the finite difference equation for the n dimensional singular oscillator can be reduced to the similar finite difference equation for the relativistic isotropic three dimensional singular oscillator we have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra

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

accurate eigenvalues and eigenfunctions for quantum mechanical anharmonic oscillators the representation of the taylor expansion of the logarithmic derivative of the wavefunction by means of a pade approximant followed by an appropriate quantization condition proves a powerful way of obtaining accurate eigenvalues of the schrodinger equation in this paper we investigate in detail some of the interesting features of this approach termed riccati pade method rpm by means of its application to anharmonic oscillators we analyse the occurrence of many roots in the neighborhoods of the physical eigenvalues in the weak coupling regime and also obtain accurate coefficients of the strong coupling expansion we finally investigate the global and the local accuracy of the rpm eigenfunctions

general recurrence and ladder relations of hypergeometric type functions a method for the explicit construction of general linear sum rules involving hypergeometric type functions and their derivatives of any order is developed this method only requires the knowledge of the coefficients of the differential equation that they satisfy namely the hypergeometric type differential equation special attention is paid to the differential recurrence or ladder relations and to the fundamental three term recurrence formulas most recurrence and ladder relations published in the literature for numerous special functions including the classical orthogonal polynomials are instances of these sum rules moreover an extension of the method to the generalized hypergeometric type functions is also described allowing us to obtain explicit ladder operators for the radial wave functions of multidimensional hydrogen like atoms where the varying parameter is the dimensionality

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