quantum algebras as quantizations of dual poisson lie groups a systematic computational approach for the explicit construction of any quantum hopf algebra uz g z starting from the lie bialgebra g that gives the first order deformation of the coproduct map z is presented the procedure is based on the well known quantum duality principle namely the fact that any quantum algebra can be viewed as the quantization of the unique poisson lie structure g g on the dual group g which is obtained by exponentiating the lie algebra g defined by the dual map from this perspective the coproduct for uz g is just the pull back of the group law for g and the poisson analogues of the quantum commutation rules for uz g are given by the unique poisson lie structure g on g whose linearization is the poisson analogue of the initial lie algebra g this approach is shown to be a very useful technical tool in order to solve the lie bialgebra quantization problem explicitly since once a lie bialgebra g is given the full dual poisson lie group g can be obtained either by applying standard poisson lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs as a consequence the quantization of g will give rise to the full uz g quantum algebra provided that ordering problems are appropriately fixed through the choice of certain local coordinates on g whose coproduct fulfils a precise quantum symmetry property the applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant lie algebras such as the 2 1 anti de sitter algebra so 2 2 and the poincare algebra in 3 1 dimensions

twisting and kappa poincare we demonstrate that the coproduct of and quantum poincare algebras in a classical algebra basis cannot be obtained by a cochain twist depending only on poincare algebra generators we also argue that the nonexistence of such a twist does not imply the nonexistence of a universal r matrix

complex quantum groups and their quantum enveloping algebras we construct complexified versions of the quantum groups associated with the lie algebras of typean 1 bn cn anddn following the ideas of faddeev reshetikhin and takhtajan we obtain the hopf algebras of regular functionals u on these complexified quantum groups in the special examplea1 we derive theq deformed enveloping algebrauq sl 2 in the limitq 1 the undeformeduq sl 2 is recovered

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

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

canonical q deformations of noncompact lie super algebras the author proposes a procedure for q deformations of the real forms g of complex lie super algebras associated with generalized cartan matrices the procedure gives different q deformations for the nonconjugate cartan subalgebras of g the author gives several illustrations e g q deformed lorentz and conformal super algebras the q deformed conformal algebra contains as a subalgebra a q deformed poincare algebra and as hopf subalgebras two conjugate 11 generator q deformed weyl algebras the q deformed lorentz algebra is a hopf subalgebra of both weyl algebras

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

harmonic analysis on spannors the spannor representation of the universal covering group of the conformal group of minkowski space time is studied herein it is an example of an indecomposable representation various parallelizations for spannors are described normalized k finite basis fields for the spannors are introduced and a computation of the actions of the scale generator and other generators of the conformal group on these basis fields is given the irreducibility of all irreducible composition factors is established and it is decided which ones are infinitesimally unitary two sets of generators for maximal abelian subalgebras of the enveloping algebra of the conformal group are introduced the action of one of the sets on k finite basis fields in the spannor representation is studied this is important for understanding of the problem of degeneracy in the spannor representation since k types occur with multiplicity 2 many calculations make use of the lowest highest weight module structures of the unitarizable and positive negative energy irreducible composition factors and thus many of our results and techniques can be used to initiate a study of the quantum deformations of the associated lie algebra representations results on the poincare content of the unitarizable irreducible composition factors are stated and unitary equivalences between certain of these factors are established

degenerate elliptic resonances quasi periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations the parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations here we prove that for a class of elliptic tori a resummation algorithm can be devised and proved to be convergent thus extending to such lower dimensional invariant tori the methods employed to prove convergence of the lindstedt series either for the maximal i e kam tori or for the hyperbolic lower dimensional invariant tori

on the structure of inhomogeneous quantum groups we investigate inhomogeneous quantum groups g built from a quantum group h and translations the corresponding commutation relations contain inhomogeneous terms under certain conditions which are satisfied in our study of quantum poincare groups 12 we prove that our construction has correct size find the r matrices and the analogues of minkowski space for g

the spurious regression of fractionally integrated processes this paper extends the theoretical analysis of the spurious regression and spurious detrending from the usual i 1 processes to the long memory fractionally integrated processes it is found that when we regress a long memory fractionally integrated process on another unrelated long memory fractionally integrated process no matter whether these processes are stationary or not as long as their orders of integration sum up to a value greater than 0 5 the t ratios become divergent and spurious effects occur our finding suggests that it is the long memory instead of nonstationarity or lack of ergodicity that causes such spurious effects as a result spurious effects might happen more often than we previously believed as they can arise even between stationary series while the usual first differencing procedure may not completely eliminate spurious effects when data possess strong long memory

quantum poincare group without dilatation and twisted classical algebra a quantum deformation of a poincare group was constructed using as a starting point su 2 2 conformal group and twistorlike definition of the minkowski space this leads to a pure twisted quantum poincare group the relative simplicity of the q deformation permits one to find out the important properties of the q groups of the space time transformations in particular it is shown that the geometrical meaning of the twisting is the transition to the another space time coordinates frame so that one can freely choose the most convenient deformation among the ones related to each other by twists the small subgroup used in the induced representation construction is proven to be twisted with respect to initial q group this last fact means that commutation relations for creation and destruction operators are not strongly defined by the q group of the space time transformations

solutions of klein gordon and dirac equations on quantum minkowski spaces covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified in the case of quantum minkowski spaces they have the same dimensions as in the classical case formal solutions of the corresponding klein gordon and dirac equations are found the fock space construction is sketched

hyperbolic low dimensional invariant tori and summations of divergent series we consider a class of a priori stable quasi integrable analytic hamiltonian systems and study the regularity of low dimensional hyperbolic invariant tori as functions of the perturbation parameter we show that under natural nonresonance conditions such tori exist and can be identified through the maxima or minima of a suitable potential they are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin the invariant tori admit an asymptotic series at the origin with taylor coefficients that grow at most as a power of a factorial and a remainder that to any order n is bounded by the n 1 st power of the argument times a power of n we show the existence of a summation criterion of the generically divergent series in powers of the perturbation size that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory

the exceptional lie algebra e7 25 multiplets and invariant differential operators in the present paper we continue the project of systematic construction of invariant differential operators on the example of the non compact exceptional algebra e7 25 our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras which we call conformal lie algebras which have very similar properties to the conformal algebras of n dimensional minkowski spacetime this class of algebras is identified and summarized in a table another motivation is related to the ads cft correspondence we give the multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

the massless representations of the conformal quantum algebra the massless representations of the conformal quantum algebra uq su 2 2 for complex q such that mod q mod 1 are studied in detail by factorizing out all singular vectors of the corresponding verma modules a simple basis for these representations is explicitly constructed this basis is new also for the usual conformal algebra su 2 2 this construction allows a straightforward treatment of the case q a root of unity when the representations are unitary and finite dimensional

borel summability and lindstedt series resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation such series may be sometimes summed via suitable sum rules defining c functions of the perturbation strength here we find sufficient conditions for the borel summability of their sums in the case of two dimensional rotation vectors with diophantine exponent 1 e g with ratio of the two independent frequencies equal to the golden mean

quantum deformation of the poincare supergroup and kappa deformed superspace the classical r matrix for the n 1 super poincare algebra given by lukierski et al 1993 is used to describe the graded poisson structure on the n 1 poincare supergroup the standard correspondence principle between the even odd poisson brackets and anti commutators leads to the consistent quantum deformation of the super poincare group with the deformation parameter q described by the fundamental mass parameter k k 1 in q the k deformation of n 1 superspace as dual to the k deformed supersymmetry algebra is discussed

non standard quantum so 2 2 and beyond a new non standard quantization of the universal enveloping algebra of the split natural real form so 2 2 of d2 is presented some classical graded contractions of so 2 2 associated to a z2 z2 grading are studied and the automorphisms defining this grading are generalized to the quantum case thus providing quantum contractions of this algebra this produces a new family of non standard quantum algebras some of these algebras can be realized as 2 1 kinematical algebras we explicitly introduce a new deformation of poincare algebra which is naturally linked to the null plane basis another realization of these quantum algebras as deformations of the conformal algebras for the two dimensional euclidean galilei and minkowski spaces is given and its new properties are emphasized

quasiperiodic attractors borel summability and the bryuno condition for strongly dissipative systems we consider a class of ordinary differential equations describing one dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping in the limit of large damping under some generic nondegeneracy condition on the force there are quasiperiodic solutions which have the same frequency vector as the forcing term we prove that such solutions are borel summable at the origin when the frequency vector is either any one dimensional number or a two dimensional vector such that the ratio of its components is an irrational number of constant type in the first case the proof given simplifies that provided in a previous work of ours we also show that in any dimension d for the existence of a quasiperiodic solution with the same frequency vector as the forcing term the standard diophantine condition can be weakened into the bryuno condition in all cases under a suitable positivity condition the quasiperiodic solution is proved to describe a local attractor

the inhomogeneous quantum groups from differential calculi with classical dimension from the bicovariant first order differential calculus on inhomogeneous hopf algebra b we construct the set of right invariant maurer cartan one forms considered as a right invariant basis of a bicovariant b bimodule over which we develop the woronowicz general theory of differential calculus on quantum groups in this formalism we introduce suitable functionals on b which control the inhomogeneous commutation rules in particular we find that the homogeneous part of commutation rules between the translations and those between the generators of the homogeneous part of b and translations are controlled by different r matrices satisfying characteristic equations

twisted conformal algebra so 4 2 a new twisted deformation uz so 4 2 of the conformal algebra of the 3 1 dimensional minkowskian spacetime is presented this construction is provided by a classical r matrix spanned by ten weyl poincare generators which generalizes non standard quantum deformations previously obtained for so 2 2 and so 3 2 however by introducing a conformal null plane basis it is found that the twist can indeed be supported by an eight dimensional carrier subalgebra by construction the weyl poincare subalgebra remains as a hopf subalgebra after deformation non relativistic limits of uz so 4 2 are shown to be well defined and they give rise to new twisted conformal algebras of galilean and carroll spacetimes furthermore a difference differential massless klein gordon or wave equation with twisted conformal symmetry is constructed through deformed momenta and position operators the deformation parameter is interpreted as the lattice step on a uniform minkowskian spacetime lattice discretized along two basic null plane directions

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

kappa deformation of poincare superalgebra with classical lorentz subalgebra and its graded bicrossproduct structure the kappa deformed d 4 poincare superalgebra written in hopf superalgebra form is transformed to the basis with classical lorentz subalgebra generators we show that in such a basis the kappa deformed d 4 poincare superalgebra can be written as a graded bicrossproduct we show that the kappa deformed d 4 superalgebra acts covariantly on a kappa deformed chiral superspace

lie algebraic structure from inhomogeneous hopf algebras we construct the vector space dual to the space of right invariant differential forms constructed from a first order differential calculus on inhomogeneous quantum groups we show that this vector space is equipped with a structure of a hopf algebra which closes on a noncommutative lie algebra satisfying a jacobi identity

quantum 2 1 kinematical algebras a global approach in this paper we give an approach to quantum deformations of the 2 1 kinematical lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in n 3 dimensions we cover at once all the kinematical geometries including the quantum versions of inonu wigner contractions which are defined in a natural way and relate q deformations as expected we thus obtain some q deformations previously known for the three dimensional euclidean and 2 1 poincare algebras and also some new q deformations for these and other kinematical algebras such as the 2 1 de sitter galilei and newton hooke algebras

four dimensional quantum affine algebras and space time q symmetries a global model of the q deformation for the quasiorthogonal lie algebras generating the groups of motions of the four dimensional affine cayley klein ck geometries is obtained starting from the three dimensional deformations it is shown how the main algebraic classical properties of the ck systems can be implemented in the quantum case quantum deformed versions either of the space time or space symmetry algebras poincare 3 1 galilei 3 1 4 d euclidean as well as others appear in this context as particular cases for some of these classical algebras several q deformations are directly obtained

lie bialgebra contractions and quantum deformations of quasi orthogonal algebras lie bialgebra contractions are introduced and classified a non degenerate coboundary bialgebra structure is implemented into all pseudo orthogonal real algebras so p q starting from the one corresponding to so n 1 it allows us to introduce a set of lie bialgebra contractions which leads to lie bialgebras of quasi orthogonal algebras this construction is explicitly given for n 2 3 4 all lie bialgebra contractions studied in this paper define hopf algebra contractions for the drinfel d jimbo deformations uzso p q they are explicitly used to generate new non semisimple quantum algebras as it is the case for the euclidean poincare and galilean algebras

quasiperiodic hamiltonian motions scale invariance harmonic oscillators the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point