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

equivalence of q bosons using the exponential phase operator various forms of the q boson are explained and their hidden symmetry revealed by transformations using the exponential phase operator both the one component and the multicomponent q bosons are discussed as a byproduct we obtain a new boson algebra having a shifted vacuum structure and define a global operator u 1 gauge transformation

isomorphisms between quantum group covariant q oscillator systems defined for q and q 1 it is shown that there exists an isomorphism between q oscillator systems covariant under suq n and suq 1 n by the isomorphism the defining relations of a suq 1 n covariant q oscillator system are transmuted into those of suq n it is also shown that a similar isomorphism exists for the system of q oscillators covariant under the quantum supergroup suq n m furthermore the cases of q deformed lie super algebras constructed from covariant q oscillator systems are considered the isomorphisms between q deformed lie super algebras cannot be obtained by the direct generalization of the isomorphism for covariant q oscillator systems

regression methods for stochastic control problems and their convergence analysis in this paper we develop several regression algorithms for solving general stochastic optimal control problems via monte carlo this type of algorithm is particularly useful for problems with a high dimensional state space and complex dependence structure of the underlying markov process with respect to some control the main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the bellman principle combined with fast methods for approximating conditional expectations and functional optimization theoretical properties of the presented algorithms are investigated and the convergence to the optimal solution is proved under some assumptions finally the presented methods are applied in a numerical example of a high dimensional controlled bermudan basket option in a financial market with a large investor

a two parameter deformation of the jaynes cummings model path integral representation a two parameter p q deformation of the jaynes cummings model is obtained using a recently developed p q deformed oscillator in the rotating wave approximation rwa the dynamical symmetry of the model is the quantum superalgebra up q 1 mod 1 the partition function of the model is obtained as a path integral over generalized perelomov coherent states corresponding to the quantum algebra up q 1 mod 1 the complete spectrum of the model is extracted for both the case when the coupling constants are grassman valued and when they are c number valued it is noted that a relaxation of the rwa extends the dynamical symmetry to the quantum superalgebra ospp q 2 mod 2

quantum deformation of bose parastatistics a q deformation of the transformation of the chevalley basis to an odd basis of generators of the universal enveloping of the lie superalgebra g n osp 1 2n is presented it is shown that one thus obtains a reasonable quantum deformation of the algebra of n para bose oscillators

new q deformed coherent states with an explicitly known resolution of unity we construct a new family of q deformed coherent states zq where 0 q 1 these states are normalizable on the whole complex plane and continuous in their label z they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem and expressed in terms of one of the two jacksons s q exponentials they also permit exact evaluation of matrix elements of physically relevant operators we use this to show that the photon number statistics for the states is sub poissonian and that they exhibit quadrature squeezing as well as an enhanced signal to quantum noise ratio over the conventional coherent state value finally we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics

the q harmonic oscillator and an analogue of the charlier polynomials a model of a q harmonic oscillator based on q charlier polynomials of al salam and carlitz 1965 is discussed a simple explicit realization of q creation and q annihilation operators q coherent states and an analogue of the fourier transformation are found a connection of the kernel of this transform with biorthogonal rational functions is observed

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 analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

generalized commutators and deformation of strong coupling superconductivity we consider quantum deformation of a strong coupling superconductivity model based on creation annihilation operators which satisfy generalized commutator relations it is found that the nature of the superconducting phase transition can be changed from the usual second order to a first order transition if the deformation parameter exceeds a certain critical value metastable normal and superconducting states can exist when the transition is of the first order

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

on closed anti de sitter spacetimes this article deals with the structure of the fundamental group of compact anti de sitter spacetimes i e lorentz manifolds with constant negative curvature algebraically such a manifold is the quotient of the universal cover of the homogeneous space so 2 n so 1 n by a discrete group gamma acting properly and co compactly on it this exists if and only if n is even indeed as this was observed by kulkarni u 1 d is contained in so 2 2d and acts properly transitively on so 2 2d so 1 2d it then suffices to take gamma as a co compact lattice in u 1 d the results of the present article give evidence to the question in dimension 3 are all compact anti de sitter spacetimes constructed in this way

extension of q deformed analysis and q deformed models of classical mechanics the author presents the generalization of the q derivative and investigates its properties the formula of conjugation for scalar products with respect to the lebesgue and discrete measures is obtained this formalism is applied to classical mechanical systems depending on functional derivatives he derives the deformed euler lagrange equations and deformed poisson bracket by an assumption that it determines the functional evolution of the systems it appears that in such models a hamiltonian analogue is no longer conserved in time for a few examples the explicit forms of constants of evolution are given

r matrix formulation of deformed boson algebra rewriting the pusz woronowicz q boson relations in terms of the standard r matrix for suq n the author gives the definition of a general deformed boson algebra a r depending upon a matrix r he investigates the conditions under which this algebra is associative for n 2 a set of matrices satisfying these conditions is classified and the corresponding twisted statistics is given

a q deformed oscillator system with quantum group slq l symmetry a q deformed oscillator system with quantum group slq l symmetry is constructed in terms of the q deformed boson operators the energy level degeneracy associated with the irreducible representation of slq l is analysed especially for the case that q is a root of unity

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

nonstandard lorentz space forms in their recent paper 8 kulharni and raymond show that a closed 3 manifold which admits a complete lorentz metric of constant curvature 1 henceforth called a complete lorentz structure must be seifert fibered over a hyperbolic base furthermore on every such seifert fibered 3 manifold with nonzero euler class they construct such a lorentz metric moreover the lorentz structure they construct has a rather strong additional property which they call standard a lorentz structure is standard if its causal double cover possesses a timelike killing vector field equivalently it possesses a riemannian metric locally isometric to a left invariant metric on sl 2 r kulkarni and raymond asked if every closed 3 dimensional lorentz structure is standard this paper provides a negative answer to this question theorem 1 and a positive answer to the implicit question raised in 8 7 1 1 theorem 3 theorem 1 let m3 be a closed 3 manifold which admits a homogeneous lorentz structure and satisfies h m r 0 then there exists a nonstandard complete lorentz structure on m in 8 it is shown that the unit tangent bundle of a closed surface admits a homogeneous lorentz structure therefore we obtain corollary 2 there exists a complete lorentz structure on the unit tangent bundle of any closed surface f of genus greater than one which is not standard the homogeneous lorentz structures are all classified in 8 a circle bundle of euler number j over a closed surface f f 0 has a homogeneous structure if and only if j f an analogous statement holds when m has singular fibers i e when f is an orbifold we also show theorem 3 let m3 be a 3 manifold which admits a complete lorentz structure then m3 is not covered by a product f x s fa closed surface theorem 3 implies that the euler class of the seifert fiber structure of m3 is nonzero

proper action on a homogeneous space of reductive type an action of l on a homogeneous space g h is investigated where l h g are reductive lie groups a criterion of the properness of this action is obtained in terms of the little weyl group of g in particular r rankg r rankh iff calabi markus phenomenon occurs i e only finite subgroups of g can act properly discontinuously on g h then by using cohomological dimension theory of a discrete group l g h is proved compact iff d g d l d h where d g denotes the dimension of a riemannian symmetric space associated with g etc these results apply to the existence problem of lattice in g h several series of classical pseudo riemannian homogeneous spaces are found to admit non uniform lattice as well as uniform lattice while some necessary condition for the existence of uniform lattice is obtained when rankg rankh partially supported by grant in aid for scientific research no 63740074

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

representations of quantum superalgebra uq gl 2 1 in a coherent state basis and generalization the coherent state method has proved to be useful in quantum physics and mathematics this method more precisely the vector coherent state method has been used by some authors to construct representations of superalgebras but almost to our knowledge it has not yet been extended to quantum superalgebras except uq osp 1 2 one of the smallest quantum superalgebras in this article the method is applied to a bigger quantum superalgebra namely uq gl 2 1 in constructing q boson fermion realizations and finite dimensional representations which when irreducible are classified into typical and nontypical representations this construction leads to a more general class of q boson fermion realizations and finite dimensional representations of uq gl 2 1 and thus at q 1 of gl 2 1 both gl 2 1 and uq gl 2 1 have found different physics applications therefore it is meaningful to construct their representations

the poisson bracket for q deformed systems it is shown that the quantum system of q deformed oscillators with the uq n group symmetry can be obtained by the canonical quantization of a classical system with a modified poisson bracket the modification of the poisson bracket is connected with a non canonical transformation of phase space variables in this approach the deformation parameter turns out to be a function of the planck constant and some dimensional parameters characterizing the classical system

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

generating function for clebsch gordan coefficients of the suq 2 quantum algebra some methods have been developed to calculate the suq 2 clebsch gordan coefficients cgc here the authors develop a method based on the calculation of clebsch gordan generating functions through the use of quantum algebraic coherent states calculating the suq 2 cgc by means of this generating function is an easy and straightforward task

completeness of coherent states associated with self similar potentials and ramanujan s integral extension of the beta function a decomposition of identity is given as a complex integral over the coherent states associated with a class of shape invariant self similar potentials there is a remarkable connection between these coherent states and ramanujan s integral extension of the beta function

realizability of a model in infinite statistics following greenberg and others we study a space with a collection of operatorsa k satisfying the q mutator relations a l a k a l k l corresponding forq 1 to classical bose and fermi statistics we show that then n matrixa n q representing the scalar products ofn particle states is positive definite for alln ifq lies between 1 and 1 so that the commutator relations have a hilbert space representation in this case this has also been proved by fivel and by bozejko and speicher we also give an explicit factorization ofa n q as a product of matrices of the form 1 q jt 1 with 1 j n andt a permutation matrix in particular a n q is singular if and only ifq m 1 for some integerm of the formk 2 k 2 k n

stratified regression monte carlo scheme for semilinear pdes and bsdes with large scale parallelization on gpus in this paper we design a novel algorithm based on least squares monte carlo lsmc in order to approximate the solution of discrete time backward stochastic differential equations bsdes our algorithm allows massive parallelization of the computations on many core processors such as graphics processing units gpus our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization in this way we minimize the exposure to the memory requirements due to the storage of simulations indeed we note the lower memory overhead of the method compared with previous works

real de sitter and conformal algebras in terms of q oscillators we express the cartan weyl basis of all quantum real d 4 de sitter and d 4 conformal algebras in terms of q deformed oscillators

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

time quantization and q deformations we extend to quantum mechanics the technique of stochastic subordination by means of which one can express any semi martingale as a time changed brownian motion as examples we considered two versions of the q deformed harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules

hedged monte carlo low variance derivative pricing with objective probabilities abstract we propose a new hedged monte carlo hmc method to price financial derivatives which allows to determine simultaneously the optimal hedge the inclusion of the optimal hedging strategy allows one to reduce the financial risk associated with option trading and for the very same reason reduces considerably the variance of our hmc scheme as compared to previous methods the explicit accounting of the hedging cost naturally converts the objective probability into the risk neutral one this allows a consistent use of purely historical time series to price derivatives and obtain their residual risk the method can be used to price a large class of exotic options including those with path dependent and early exercise features

nonlinear lie algebra and ladder operators for orbital angular momentum based on the nonlinear lie algebra a new framework is put forward to treat the problem of angular momentum from this framework the ladder operators for the orbital angular momentum can be solved as simultaneous eigenvectors of the lie derivations ad l2 and ad l2 in an operator space the ladder operators for solvable central potentials can also be found in a united way

q to or from q 1 invariance of q oscillators and new realizations of quantum algebras new realizations of quantum algebras suq 2 and suq 1 1 are constructed in terms of the q bosonic and the q fermionic oscillators they satisfy the defining commutation relations of quantum algebras without giving any new structure on the q oscillators a new relation between the q oscillators defined for q and those for q 1 is found it converts the defining relations of the q oscillators for q 1 to those for q using this relation it can be shown that these realizations are invariant under q to or from u 1

multi parametric deformed heisenberg algebras a route to complexity we introduce a generalization of the heisenberg algebra which is written in terms of a functional of one generator of the algebra f j0 that can be any analytical function when f is linear with slope we show that the algebra in this case corresponds to q oscillators for q2 tan the case where f is a polynomial of order n in j0 corresponds to an n parameter deformed heisenberg algebra the representations of the algebra when f is any analytical function are shown to be obtained through the study of the stability of the fixed points of f and their composed functions the case when f is a quadratic polynomial in j0 the simplest nonlinear scheme which is able to create chaotic behaviour is analysed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of heisenberg algebra

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

generalized q oscillators and their hopf structures we study the relationships among the various forms of the q oscillator algebra and consider the conditions under which it supports a hopf structure we also present a generalization of this algebra together with its corresponding hopf structure its multimode extensions are also considered

three dimensional quantum groups from contractions of su 2 q contractions of lie algebras and of their representations are generalized to define new quantum groups an explicit and complete exposition is made for the one dimensional heisenberg h 1 q and the two dimensional euclidean quantum group e 2 q obtained by contracting su 2 q

three dimensional quadratic algebras some realizations and representations four classes of three dimensional quadratic algebra of the type q0 q q q q aq02 bq0 c where a b c are constants or central elements of the algebra are constructed using a generalization of the well known two mode bosonic realizations of su 2 and su 1 1 the resulting matrix representations and single variable differential operator realizations are obtained some remarks on the mathematical and physical relevance of such algebras are given

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

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

evolution operators for quantum chains discrete time evolution operators in integrable quantum lattice models are sometimes more fundamental objects than hamiltonians in this paper we study an evolution operator for the one dimensional integrable q deformed bose gas with xxz type impurities and find its spectrum evolution operators give a new interpretation of known integrable systems for instance our system describes apparently a simplest laser with a clear resonance peak in the spectrum

brst cohomology for uq sl 2 representations a nilpotent brst operator qq is constructed for the q deformation of the universal enveloping algebra of sl 2 an associated operator rq such that qq rq cq is introduced where cq is the q deformed casimir invariant cq fe 1 2h q 1 2h 1 q the related operator qq conjugate to qq is also considered the resulting brst cohomology of uq sl 2 representations is discussed

the distribution of zeros of general q polynomials a general system of q orthogonal polynomials is defined by means of its three term recurrence relation this system encompasses many of the known families of q polynomials among them the q analogue of the classical orthogonal polynomials the asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation this result is applied to specific classes of polynomials known by the names q hahn q kravchuk q racah q askey and wilson al salam carlitz and the celebrated little and big q jacobi

approach of spherical harmonics to the representation of the deformed su 1 1 algebra the m shifting generators of su 2 algebra together with a pair of l shifting ladder symmetry operators have been used in the space of all spherical harmonics ylm in order to introduce a new set of operators expressing the transitions between them it is shown that the space of spherical harmonics whose l 2m or l 2m is given presents negative and positive irreducible representations of a deformed su 1 1 algebra respectively these internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum generating algebras

a q analog of the quantum central limit theorem for suq 2 a q analog of the central limit theorem for suq 2 q 0 is studied it is shown that the limits of the moments and q exponential generating functions for coherent states give for q 1 0 q 1 the harmonic q oscillator q 1 oscillator introduced by biedenharn and macfarlane

a study of representations of the algebra of functions on the quantum group glq n a q analogue of the root system is constructed for this algebra which is similar to the root system of classical lie algebras it is then used to construct in detail a class of representations of the algebra of functions on the quantum group glq n and a q boson realization of the generators of glq n is given i also construct infinite dimensional hilbert space representations of this algebra the main result of this paper is stated in proposition 8

suq h cross to 0 2 and suq h cross 2 the classical and quantum q deformations of the su 2 algebra ii the hopf algebra the yang baxter equation and multi deformed algebraic structures for pt i see ibid vol 23 p 4185 1990 the suq 2 algebra is realized by means of both the poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q deformed oscillators of two different types the structures of the hopf algebra and the quantum yang baxter equation are also discussed on a quantum level a set of j representations of the quantum algebra suq 2 is constructed based on the type ii q oscillators multi deformations of the oscillators of the two types and multi deformed algebras expressed in poisson brackets as well as in lie brackets are proposed

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

a nonintrusive stratified resampler for regression monte carlo application to solving nonlinear equations our goal is to solve certain dynamic programming equations associated to a given markov chain x using a regression based monte carlo algorithm more specifically we assume that the model for x is not known in full detail and only a root sample x1 xm of such process is available by a stratification of the space and a suitable choice of a probability measure we design a new resampling scheme that allows to compute local regressions on basis functions in each stratum the combination of the stratification and the resampling allows to compute the solution to the dynamic programming equation possibly in large dimensions using only a relatively small set of root paths to assess the accuracy of the algorithm we establish non asymptotic error estimates in l2 our numerical experiments illustrate the good performance even with m 20 40 root paths

on the q symplecton realization of the quantum group suq 2 a q symplecton realization of the quantum group suq 2 is constructed giving explicitly the product law for the weyl ordered characteristic polynomials a new form of the q racah coefficients is obtained which is especially adapted to proving a new symmetry for these functions a surprising result is that unlike the su 2 case there are two distinct invariant q triangle functions in the q symplecton extension

verma module of the quantum group gl n q and its q boson and heisenberg weyl relation realizations the structure and verma module of the quantum matrix element algebra a n q of the quantum group gl n q are studied the method used here is similar to that for studying the structure and verma module of semisimple lie algebras the concept of the cartan subalgebra raising and lowering operators and their pairs is defined the q boson and the corresponding heisenberg weyl relation realizations are generally studied and the cyclic representations of a n q are obtained the explicit examples a 2 q and a 3 q are discussed in detail

algebraic structure of parabose fock space i the green s ansatz revisited a description of the fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed a new statement of the well known green s ansatz is introduced to help understand the significance of the bilinear commutation relations it is also applied to show how all parabose oscillators of even order respectively odd order can be obtained as irreducible constituents of the fock space associated with a green s ansatz of two three terms analogs of the form in which the parabose green s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order methods used in the paper allow various new results for q deformed parabose oscillators to be derived

path integral formalism for suq 2 coherent states a path integral formulation in the representation of coherent states for the quantum group suq 2 is introduced an expression for the transition amplitude connecting two suq 2 coherent states is constructed and the corresponding canonical equations of motion are also derived

a deformed oscillator with coulomb energy spectrum a deformed oscillator with eigenvalues equal to the eigenvalues of the schrodinger equation with the coulomb potential is constructed the deformed oscillator algebra has a polynomial realization and the associate deformed operations of integration and differentiation are studied

a new completeness relation in the q deformed two mode fock space in two mode fock space a new completeness relation for the q analogue of the single complex variable state is proved by virtue of some q deformed combinatorial identities which are also derived here for the first time

on position and momentum operators in the q oscillator algebra the aim of this paper is to study the position and momentum operators in q deformed oscillator algebras the natural form of the position operator is xp qpn a a qpn where p is a real number this operator is an operator representable by a jacobi matrix using the theory of jacobi matrices the theory of classical moment problem and the theory of basic hypergeometric functions it is shown that depending on values of q and p xp can be unbounded symmetric operator which has the deficiency indices 1 1 and hence is not self adjoint but has self adjoint extensions bounded self adjoint operator with continuous simple spectrum or self adjoint operator of trace class therefore with discrete spectrum with zero as the point of accumulation of eigenvalues the connection of the q deformed heisenberg relation px qxp 1 for the position and momentum operators with a q deformation of the quantum harmonic oscillator is also considered

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

q oscillators realization of fq 4 and gq 3 a realization of the quantum superalgebras corresponding to f 4 and g 3 is given in terms of q deformed bosonic and fermionic oscillators

unitarizable representations of the deformed para bose superalgebra uq osp 1 2 at roots of 1 the unitarizable irreps of the deformed para bose superalgebra pbq which is isomorphic to uq osp 1 2 are classified at q being root of 1 new finite dimensional irreps of uq osp 1 2 are found explicit expressions for the matrix elements are written down

the q boson realization of parametrized cyclic representations of quantum algebras at qp 1 a parametrized cyclic representation of the q deformed heisenberg weyl algebras is constructed the cyclic representations of any quantum algebras are studied in terms of their q boson realizations a general method to construct the q boson realizations of quantum algebras from their verma representations is proposed two explicit examples sl 2 q and sl 3 q are studied in detail some cyclic representations of quantum algebras sl n q and uqcn are presented by using their q boson realizations

the q boson realizations of the quantum group uq sl n 1 c the authors give an explicit expression of the recurrency formulae of canonical realization for quantum enveloping algebras uq sl n 1 c in these formulae the generators of the algebra uq sl n 1 c are expressed by means of n canonical q boson pairs one auxiliary representation of the algebra uq gl n c

jacobson generators fock representations and statistics of sl n 1 the properties of a statistics related to the class a of simple lie algebras t d palev preprint jinr e17 10550 1977 hep th 9705032 are further investigated the description of each sl n 1 is carried out via generators a1 an which we call jacobson generators with respect to these generators the definition of a fock space of sl n 1 is given it is proved that the fock spaces wp p n are the simple symmetric finite dimensional modules of sl n 1 the pauli principle of the underlying statistics is formulated within each wp operators b p i ai p i 1 n called quasi bose creation and annihilation operators caos are defined then limp b p i bi where bi are ordinary bose caos therefore a statistics appears as an approximation of bose statistics with caos acting in finite dimensional state spaces we indicate that the p 1 quasi bose operators b 1 1 b 1 n are natural operators for the description of hard core bose models and of the related heisenberg spin models we argue th

a q analogue of the supersymmetric oscillator and its q superalgebra a q analogue of the supersymmetric oscillator is constructed out of q boson and q fermion creation and annihilation operators for a fixed nb nf 2n where nb and nf are q boson and q fermion occupation numbers the irreducible representations of the q superalgebra generated by the q oscillator are 2n 1 dimensional particular cases when q is a root of or 1 are discussed a realization of the quantum group suq 2 is obtained using a pair of 1 fermion creation and annihilation operators

on quantum deformations of the simplest lie parasuperalgebra quantum deformations of the simplest lie parasuperalgebra psqm 2 are constructed by taking advantage of the well known representations of suq 2 c as a particular result the authors show that the fundamental witten superalgebra sqm 2 cannot be deformed

integrable representations of the ultra commutation relations we consider representations of the unital complex algebra generated by the identity and two elements and with and one relation the ultra commutation relations ucr in general we do not impose any commutation relation between and this is a very general scheme encompassing many important physical examples inter alia the ccr car q deformed bosons and fermions the representations of interest in physics have a diagonal number operator whose spectrum is contained in the positive integers together with some other technical conditions our principal result is that every representation in this class is completely determined up to unitary equivalence by the sequence of numbers n 1 n 1 n 2 for n 0 with 0 0 here n is the normalized eigenvector of corresponding to the eigenvalue n if the dimension of that eigenspace is 1 if the carrier hilbert space is infinite dimensional this representation is irreducible if and only if n 0 for n 1 finall

hopf superalgebra contractions and r matrix for fermions different contractions are applied to the quantum superalgebra ospq 1 mod 2 in the first of them the graded analogue s hq 1 of the one dimensional heisenberg quantum algebra is obtained and its r matrix explicitly calculated in the second contraction a z2 graded version of the q oscillator is proposed and finally the supersymmetric two dimensional euclidean quantum algebra s eq 2 is found

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

the fundamental group of a compact flat lorentz space form is virtually polycyclic a flat lorentz space form is a geodesically complete lorentzian manifold of zero curvature it is well known see auslander markus 3 that such a space m may be represented as a quotient r where r is an dimensional minkowski space n equals the dimension of m and is a group of lorentz isometries acting properly discontinuously and freely on r in particular the universal covering of m is isometric to r and the fundamental group m is isomorphic to theorem let m be a compact flat lorentz space form then m is virtually poly cyclic recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups this result affirms a conjecture of milnor 13 in a special case for discussion of this conjecture as well as another special case we refer to fried goldman 8 the importance of this result is that it reduces the classification of compact flat lorentz space forms to fairly elementary problems concerning lie algebras and lattices in solvable lie groups we hope to pursue this classification in a future publication for a description of this reduction and the classification in dimension 3 see fried goldman 8 in dimension 4 the classification is worked out in fried 7 one immediate consequence of the structure theory developed in fried goldman 8 1 and kamishima 12 is the following corollary let m be the compact flat lorentz space form then m has a finite covering which is diffeomorphic to a solvmanifold the outline of this paper is as follows in the first section we collect some basic facts about the group e n 1 1 of isometries of dimensional minkowski space in the second section we prove the theorem in the special

generalized heisenberg algebra application to the harmonic oscillator the deformed poisson algebra recently introduced to investigate integrable systems 2003 j phys a math gen 36 12181 203 2005 j math phys 46 042702 is used to perform the transition from the phase space of classical observables functions depending on positions and momentums to the hilbert space of physically well defined hermitian operators a hamiltonian operator for the harmonic oscillator system is constructed and the eigenvalue problem is solved the generalization to an n dimensional space shows that such an algebra does not break the rotational symmetry

on a q generalization of circular and hyperbolic functions a generalization of the circular and hyperbolic functions is proposed based on the tsallis statistics and on a consistent generalization of the euler formula some properties of the presently proposed q trigonometry are then investigated the generalized functions are exact solutions of a nonlinear oscillator original circular and hyperbolic relations are recovered as the limiting case

an algebraic q deformed form for shape invariant systems a quantum deformed theory applicable to all shape invariant bound state systems is introduced by defining q deformed ladder operators we show that these new ladder operators satisfy new q deformed commutation relations in this context we construct an alternative q deformed model that preserves the shape invariance property presented by the primary system q deformed generalizations of morse scarf and coulomb potentials are given as examples

the q supercoherent states of the q deformed su 2 superalgebra the q supercoherent states associated with the q deformed su 2 superalgebra are constructed explicitly and their properties are investigated in detail the completeness relation of the q supercoherent states is proved by the use of the q integration

alternative hamiltonian quantization the authors take a standard hamiltonian quadratic in a set of creation and annihilation operators and demonstrate an alternative quantization by quommutators i e a set of rules for transposition of these operators different from the those of the usual canonical commutation relations which yields a new spectrum of a type reminiscent of logarithmic trajectories