peripheric extended twists the properties of the set of extended jordanian twists are studied it is shown that the boundaries of contain twists whose characteristics differ considerably from those of internal points the extension multipliers of these peripheric twists are factorizable this leads to simplifications in the twisted algebra relations and helps to find the explicit form for coproducts the peripheric twisted algebra u sl 4 is obtained to illustrate the construction it is shown that the corresponding deformation up sl 4 cannot be connected with the drinfeld jimbo one by a smooth limit procedure all the carrier algebras for the extended and the peripheric extended twists are proved to be frobenius

chains of frobenius subalgebras of so m and the corresponding twists chains of extended jordanian twists are studied for the universal enveloping algebras u so m the carrier subalgebra of a canonical chain fb0 pmax cannot cover the maximal nilpotent subalgebra n so m we demonstrate that there exist other types of frobenius subalgebras in so m that can be large enough to include n so m the problem is that the canonical chains fb0 p do not preserve the primitivity on these new carrier spaces we show that this difficulty can be overcome and the primitivity can be restored if one changes the basis and passes to the deformed carrier spaces finally the twisting elements for the new frobenius subalgebras are explicitly constructed this gives rise to a new family of universal r matrices for orthogonal algebras for a special case of g so 5 and its defining representation we present the corresponding matrix solution of the yang baxter equation

non standard quantum 1 1 poincare group a t matrix approach the hopf algebra dual form for the non standard uniparametric deformation of the 1 1 poincare algebra iso 1 1 is deduced in this framework the quantum coordinates that generate funw iso 1 1 define an infinite dimensional lie algebra the t matrix formalism is used to derive a universal r matrix for both uwiso 1 1 and funw iso 1 1 it is also shown how these results can be generalized for the triangular deformations of 1 1 poincare and galilei algebras that include a spacetime dilation generator

symbolic diffusion entropy rate of chaotic time series as a surrogate measure for the largest lyapunov exponent existing methods for estimating the largest lyapunov exponent from a time series rely on the rate of separation of initially nearby trajectories reconstructed from the time series in phase space according to ueda chaotic dynamical behavior is viewed as a manifestation of random transitions between unstable periodic orbits in a chaotic attractor which are triggered by perturbations due to experimental observation or the roundoff error characteristic of the computing machine and consequently consists of a sequence of piecewise deterministic processes instead of an entirely deterministic process chaotic trajectories might have no physical reality here we propose a mathematical method for estimating a surrogate measure for the largest lyapunov exponent on the basis of the random diffusion of the symbols generated from a time series in a chaotic attractor without resorting to initially nearby trajectories we apply the proposed method to numerical time series generated by chaotic flow models and verify its validity

a quantum duality principle for coisotropic subgroups and poisson quotients we develop a quantum duality principle for coisotropic subgroups of a formal poisson group and its dual namely starting from a quantum coisotropic subgroup for a quantization of a given poisson group we provide functorial recipes to produce quantizations of the dual coisotropic subgroup in the dual formal poisson group by the natural link between subgroups and homogeneous spaces we argue a quantum duality principle for poisson homogeneous spaces which are poisson quotients i e have at least one zero dimensional symplectic leaf as an application we provide an explicit quantization of the homogeneous sln space of stokes matrices with the poisson structure given by dubrovin and ugaglia

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

the three dimensional euclidean quantum group e 3 q and its r matrix a contraction procedure starting from so 4 q is used to determine the quantum analog e 3 q of the three dimensional euclidean group and the structure of its representations a detailed analysis of the contraction of the r matrix is then performed and its explicit expression has been found the classical limit of r is shown to produce an integrable dynamical system by means of the r matrix the pseudogroup of the noncommutative representative functions is considered it will finally be shown that a further contraction made on e 3 q produces the two dimensional galilei quantum group and this in turn can be used to give a new realization of e 3 q and e 2 1 q

drinfel d superdoubles and poisson lie t plurality in low dimensions defining the real lie superalgebra as real z2 graded vector space we classify real manin supertriples and drinfel d superdoubles of superdimensions 2 2 4 2 and 2 4 the drinfel d doubles of the superdimension 2 2 are then used for construction of the simplest models related by poisson lie t plurality

behavior of solutions near manifolds of periodic solutions abstract consider an autonomous differential system x f x of dimension n that admits a k dimensional invariant manifold in x space represented by a k parameter family of periodic solutions then k characteristic multipliers of the corresponding variational equation are equal to 1 under the hypothesis that none of the remaining n k characteristic multipliers has modulus 1 the behavior of the solutions of x f x near is investigated the main results are the description of a neighborhood of in terms of stable and unstable manifolds and the statement that whenever a solution of x f x approaches the invariant manifold as t it tends to with asymptotic amplitude and phase

dispersive graded entropy on computing dynamical complexity abstract we propose a phase space based statistical disorder to investigate the dynamical complexity of chaotic models the statistical disorder is defined by introducing a grade function inversely maps the mean dispersion of the trajectories in the phase space we denote the associate entropy by the dispersive graded entropy dge numerical investigation shows that dge can quantify the dynamical complexity of discrete as well as continuous chaotic systems a comparative study is also made with the other phase space based entropy measures finally the proposed measure has been applied on three types of heart rate variability hrv signals the results support the clinical observations related to the dynamics of healthy and congestive hearts

quantum e 2 groups and lie bialgebra structures lie bialgebra structures on e 2 are classified for two lie bialgebra structures which are not coboundaries i e those which are not determined by a classical r matrix we solve the cocycle condition find the lie poisson brackets and obtain quantum group relations there is one to one correspondence between lie bialgebra structures on e 2 and possible quantum deformations of u e 2 and e 2

on quantization of z2 graded algebras quantization of z2 graded lie bialgebras viewed as a generalization of sl 2 is investigated a method of building multiparameter solutions is developed we study the role of the quantum double construction in the selected class and prove that a quantized coboundary z2 graded lie bialgebra contains a quasitriangular hopf subalgebra

extended and reshetikhin twists for sl 3 the properties of the set of extended jordanian twists for algebra sl 3 are studied starting from the simplest algebraic construction the peripheric hopf algebra we explicitly construct the complete family of extended twisted algebras sl 3 corresponding to the set of four dimensional frobenius subalgebras l in sl 3 it is proved that the extended twisted algebras with different values of the parameter are connected by a special kind of reshetikhin twist we study the relations between the family sl 3 and the one dimensional set sl 3 produced by the standard reshetikhin twist from the drinfeld jimbo quantization sl 3 these sets of deformations are in one to one correspondence each element of sl 3 can be obtained by a limiting procedure from the unique point in the set sl 3

generalized quantization scheme for lie algebras the generalized form of the coproduct for quantum lie algebras is proposed it is demonstrated that the admissible quantum compositions can be easily found the whole construction gives rise to the wide class of multidimensional hopf algebras the efficiency of the method is demonstrated on examples of nonsemisimple quantum universal enveloping algebras

quantization of coboundary lie bialgebras we show that any coboundary lie bialgebra can be quantized for this we prove that a etingof kazhdan quantization functors are compatible with lie bialgebra twists and b if such a quantization functor corresponds to an even associator then it is also compatible with the operation of taking coopposites we also use the relation between the etingof kazhdan construction of quantization functors and the alternative approach to this problem which was established in a previous work

convergence of nelson fiffusions let t t n n 1 be solutions of schrodinger equations with potentials form bounded by 1 2 and initial data inh1 d letp p n n 1 be the probability measures on the path space c d given by the corresponding nelson diffusions we show that if t n n 1 converges to t inh1 d uniformly int over compact intervals then open image in new window converges to open image in new window in total variation t 0 moreover if the potentials are in the kato classk d we show that the above result follows fromh1 convergence of initial data andk d convergence of potentials

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

an r matrix approach to the quantization of the euclidean group e 2 the r matrices for two different deformations of the euclidean group e 2 calculated in a two dimensional representation are used to determine the deformed hopf algebra of the representative functions the duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified

classification of three dimensional lie bialgebras the paper contains a classification of the three dimensional real and complex lie bialgebras with a description of the corresponding drinfel d doubles

the drinfeld double gl n tn we present a self dual drinfeld double structure underlying the an series of simple lie algebras such double is constructed through a central extension tn of gl n and is obtained by pairing two disjoint solvable subalgebras coming from positive and negative roots the cartan weyl basis of gl n is shown to be completely determined by the compatibility conditions in the double a natural lie bialgebra structure on gl n is obtained that offers a new perspective for the construction of its quantum deformations

exotic quantum double its universal r matrix and their representations the exotic quantum double and its universal r matrix for the quantum yang baxter equation are constructed in terms of drinfeld s quantum double theory as a new quasitriangular hopf algebra it is different from those standard quantum doubles that are the q deformations for lie algebras or lie superalgebras by studying its representation theory many parameter representations of the exotic quantum double are obtained with an explicit example associated with lie algebra a2 the multiparameter r matrices for the quantum yang baxter equation can result from the universal r matrix of this exotic quantum double and these representations

kappa minkowski spacetime kappa poincare hopf algebra and realizations we unify minkowki spacetime and lorentz algebra in unique lie algebra introducing commutative momenta a family of deformed heisenberg algebras and deformed poincare algebras are defined they are specified by the matrix depending on momenta we construct all such matrices realizations and star product are defined and analyzed in general and specially their relation to coproduct of momenta is pointed out hopf algebra of the poincare algebra related to the covariant realization is presented in unified covariant form left right dual realizations and dual algebra are introduced and considered the generalized involution and the star inner product are analyzed and their properties are discussed partial integration and deformed trace property are obtained in general the translation invariance of the star product is pointed out finally perturbative approach up to the first order in a is presented in the appendix

the classical basis for the κ poincare hopf algebra and doubly special relativity theories several issues concerning the quantum poincare algebra are discussed and reconsidered here we propose two different formulations of the poincare quantum algebra firstly we present a complete hopf algebra formula of poincare in classical poincare basis further by adding one extra generator which modifies the classical structure of the poincare algebra we eliminate nonpolynomial functions in the parameter hilbert space representations of such algebras make doubly special relativity dsr similar to stueckelberg s version of proper time relativistic quantum mechanics

quantum 1 1 extended galilei algebras from lie bialgebras to quantum r matrices and integrable systems the lie bialgebras of the 1 1 extended galilei algebra are obtained and classified into four multiparametric families their quantum deformations are obtained together with the corresponding deformed casimir operators for the coboundary cases quantum universal r matrices are also given applications of the quantum extended galilei algebras to classical integrable systems are explicitly developed

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 quantum algebras a cartan like point of view a perturbative quantization procedure for lie bialgebras is introduced the relevance of the choice of a completely symmetrized basis of the quantum universal enveloping algebra is stressed sets of elements of the quantum algebra that play a role similar to generators in the case of lie algebras are considered and a cartan like procedure applied to find a representative for each class of quantum algebras the method is used to construct and classify all three dimensional complex quantum algebras that are compatible with a given type of coproduct the quantization of all lie algebras that in the classical limit belong to the most relevant sector in the classification for three dimensional lie bialgebras given in 1 is thus performed new quantizations of solvable algebras whose simplicity makes them suitable for possible physical applications are obtained and already known related quantum algebras recovered

rtt relations a modified braid equation and noncommutative planes with the known group relations for the elements a b c d of a quantum matrix t as input a general solution of the rtt relations is sought without imposing the yang baxter yb constraint for r or the braid equation for r pr for three biparametric deformatios gl p q 2 gl g h 2 and gl q h 1 1 the standard the nonstandard and the hybrid one respectively r or r is found to depend apart from the two parameters defining the deformation in question on an extra free parameter k such that r 12 r 23 r 12 r 23 r 12 r 23 k k1 1 k k2 1 r 23 r 12 with k1 k2 1 p q 1 1 and 1 1 q respectively only for k k1 or k k2 one has the braid equation arbitray k corresponds to a class conserving the group relations independent of k of the mqybe or modified quantum yb equations studied by gerstenhaber giaquinto and schack various properties of the triparametric r k p q r k g h and r k q h are studied in the larger space of the modified braid equation mbe even r k p

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

chains of twists for classical lie algebras for chains of regular injections of hopf algebras the sets of maximal extended jordanian twists k are considered we prove that under certain conditions there exists for 0 the twist composed of the factors k the general construction of a chain of twists is applied to the universal envelopings u of classical lie algebras we study the chains for the infinite series an bn and dn the properties of the deformation produced by a chain u are explicitly demonstrated for the case of so 9

quaternionic and poisson lie structures in three dimensional gravity the cosmological constant as deformation parameter each of the local isometry groups arising in three dimensional 3d gravity can be viewed as a group of unit split quaternions over a ring which depends on the cosmological constant in this paper we explain and prove this statement and use it as a unifying framework for studying poisson structures associated with the local isometry groups we show that in all cases except for the case of euclidean signature with positive cosmological constant the local isometry groups are equipped with the poisson lie structure of a classical double we calculate the dressing action of the factor groups on each other and find among others a simple and unified description of the symplectic leaves of su 2 and sl 2 r we also compute the poisson structure on the dual poisson lie groups of the local isometry groups and on their heisenberg doubles together they determine the poisson structure of the phase space of 3d gravity in the so called combinatorial description

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

n 2 structures on solvable lie algebras the c 9 classification let mathfrak g be a finite dimensional lie algebra not necessarily semisimple it is known that if mathfrak g is self dual that is if it possesses an invariant metric then it admits ann 1 affine sugawara construction under certain additional hypotheses thisn 1 structure admits ann 2 extension if this is the case mathfrak g is said to possess ann 2 structure it is also known that ann 2 structure on a self dual lie algebra mathfrak g is equivalent to a vector space decomposition mathfrak g mathfrak g oplus mathfrak g where mathfrak g pm are isotropic lie subalgebras in other words n 2 structures on mathfrak g in one to one correspondence with manin triples mathfrak g mathfrak g mathfrak g in this paper we exploit this correspondence to obtain a classification of thec 9n 2 structures on solvable lie algebras in the process we also give some simple proofs for a variety of lie algebras in the process we also give some simple proofs for a variety of lie algebraic results concerning self dual lie algebras admitting symplectic or kahler structures

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

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

quantization of lie poisson structures by peripheric chains the quantization properties of composite peripheric twists are studied peripheric chains of extended twists are constructed for u sl n in order to obtain composite twists with sufficiently large carrier subalgebras it is proved that the peripheric chains can be enlarged with additional reshetikhin and jordanian factors this provides the possibility to construct new solutions to drinfeld equations and thus to quantize new sets of lie poisson structures when the jordanian additional factors are used the carrier algebras of the enlarged peripheric chains are transformed into algebras of motion of the form the factor algebra is a direct sum of borel and contracted borel subalgebras of lower dimensions the corresponding form is a coboundary the enlarged peripheric chains represent the twists that contain operators external with respect to the lie poisson structure the properties of new twists are illustrated by quantizing r matrices for the algebras u sl 3 u sl 4 and u sl 7

poisson lie groups bi hamiltonian systems and integrable deformations given a lie poisson completely integrable bi hamiltonian system on we present a method which allows us to construct under certain conditions a completely integrable bi hamiltonian deformation of the initial lie poisson system on a non abelian poisson lie group of dimension n where is the deformation parameter moreover we show that from the two multiplicative poisson lie hamiltonian structures on that underly the dynamics of the deformed system and by making use of the group law on one may obtain two completely integrable hamiltonian systems on by construction both systems admit reduction via the multiplication in to the deformed bi hamiltonian system in the previous approach is applied to two relevant lie poisson completely integrable bi hamiltonian systems the lorenz and euler top systems

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

quantum jordanian twist the quantum deformation of the jordanian twist q for the standard quantum borel algebra uq b is constructed it gives the family uq b of quantum algebras depending on parameters and h in a generic point these algebras represent the hybrid standard nonstandard quantization the quantum jordanian twist can be applied to the standard quantization of any kac moody algebra the corresponding classical r matrix is a linear combination of the drinfeld jimbo and the jordanian ones the two parametric families of hopf algebras obtained here are smooth and for the limit values of the parameters the standard and nonstandard quantizations are recovered the twisting element q also has correlated limits in particular when q tends to unity it acquires the canonical form of the jordanian twist to illustrate the properties of the quantum jordanian twist we construct the hybrid quantizations for u sl 2 and for the corresponding affine algebra the universal quantum matrix and its defining representation are presented

basic twist quantization of the exceptional lie algebra g2 we present the formulas for twist quantization of g2 corresponding to the solution of a classical yb equation with support in the eight dimensional borel subalgebra of g2 the considered chain of twists consists of the four factors describing the four steps of quantization jordanian twist the two twist factors extending jordanian twist and the deformed jordanian or in a second variant additional abelian twist the first two steps describe as well the sl 3 quantization the coproducts are calculated for each step in explicit form and for that purpose we present new formulas for the calculation of similarity transformations on the tensor product we introduce new basic generators in universal enveloping algebra u g2 that provide nonlinearities in the algebraic sector maximally simplifying the deformed coproducts

a new quasitriangular hopf algebra as the nontrivial quantum double of a simplest c algebra and its universal r matrix for yang baxter equation endowing a simple c algebra generated by two commuting elements and a unit with a noncocommutative hopf algebra structure a new quantum double and the corresponding universal r matrix for the yang baxter equation is constructed the finite dimensional representations of the quantum double is also studied and a concrete r matrix is presented as an example

from quantum universal enveloping algebras to quantum algebras the local structure of a quantum group gq is currently considered to be an infinite dimensional object the corresponding quantum universal enveloping algebra uq g which is a hopf algebra deformation of the universal enveloping algebra of an n dimensional lie algebra g lie g however we show how by starting from the generators of the underlying lie bialgebra g the analyticity in the deformation parameter s allows us to determine in a unique way a set of n almost primitive basic objects in uq g which could be properly called the quantum algebra generators so the analytical prolongation gq of the lie bialgebra g is proposed as the appropriate local structure of gq besides as in this way g and uq g are shown to be in one to one correspondence the classification of quantum groups is reduced to the classification of lie bialgebras the suq 2 and suq 3 cases are explicitly elaborated

h k trichotomies and asymptotics of nonautonomous difference systems abstract discrete nonautonomous nonlinear systems possessing h k trichotomies are considered we specialize to the case of solutions moving inside and in a neighborhood of an invariant manifold of a special kind which we call h k hyperbolic the aulbach coppel knobloch transformation is then developed and used to state conditions for granting asymptotic equivalence of solutions our results generalize some of aulbach s results in which the manifold considered was made of stationary points and the system taken was autonomous

1 1 schrodinger lie bialgebras and their poisson lie groups all lie bialgebra structures for the 1 1 dimensional centrally extended schrodinger algebra are explicitly derived and proved to be of coboundary type therefore since all of them come from a classical r matrix the complete family of schrodinger poisson lie groups can be deduced by means of the sklyanin bracket all possible embeddings of the harmonic oscillator extended galilei and gl 2 lie bialgebras within the schrodinger classification are studied as an application new quantum hopf algebra deformations of the schrodinger algebra including their corresponding quantum universal r matrices are constructed

lie bi algebra structures for centrally extended two dimensional galilei algebra and their lie poisson counterparts all bi algebra structures for centrally extended galilei algebra are classified the corresponding lie poisson structures on centrally extended galilei group are found

invariant manifolds with asymptotic phase for nonautonomous difference equations abstract for autonomous difference equations with an invariant manifold conditions are known which guarantee that a solution approaching this manifold eventually behaves like a solution on this manifold in this paper we extend the fundamental result in this context to difference equations which are nonautonomous and whose solutions are guaranteed only in forward time

deformed minkowski spaces classification and properties using general but simple covariance arguments we classify the quantum minkowski spaces for dimensionless deformation parameters this requires a previous analysis of the associated lorentz groups which reproduces a previous classification by woronowicz and zakrzewski as a consequence of the unified analysis presented we give the commutation properties the deformed and central length element and the metric tensor for the different spacetime algebras

generalization of the h deformation to higher dimensions we construct glh 3 from glq 3 by a singular map we show that there exist two singular maps which map glq 3 to new quantum groups we also construct their r matrices and will show that although the maps are singular their r matrices are not then we generalize these singular maps to the case gl n and for cn series

the gervais neveu felder equation for the jordanian quasi hopf uh y sl 2 algebra using a contraction procedure we construct a twist operator that satisfies a shifted cocycle condition and leads to the jordanian quasi hopf uh y sl 2 algebra the corresponding universal h y matrix obeys a gervais neveu felder equation associated with the uh y sl 2 algebra for a class of representations the dynamical yang baxter equation may be expressed as a compatibility condition for the algebra of the lax operators

quantization of drinfel d doubles hopf algebra quantizations of four dimensional and six dimensional real classical drinfel d doubles are studied by following a direct analytic approach and the full quantization is explicitly obtained for most of them several new four and six dimensional quantum algebras are presented and some general features of the method are emphasized

heisenberg xxz model and quantum galilei group the 1d heisenberg spin model with an anisotropy of the xxz type is analysed in terms of the symmetry given by the quantum galilei group gamma q 1 for a chain with an infinite number of sites the authors show that the magnon excitations and the s 1 2 n magnon bound states are determined by the algebra in this case the gamma q 1 symmetry provides a description naturally compatible with the bethe ansatz the recurrence relations determined by gamma q 1 permit one to express the energy of the n magnon bound states in a closed form in terms of tchebischeff polynomials

quantum and braided linear algebra quantum matrices a r are known for every r matrix obeying the quantum yang baxter equations it is also known that these act on vectors given by the corresponding zamalodchikov algebra this interpretation is developed in detail distinguishing between two forms of this algebra v r vectors and v r covectors a r v r21 v r is an algebra homomorphism i e quantum matrices are realized by the tensor product of a quantum vector with a quantum covector while the inner product of a quantum covector with a quantum vector transforms as a scaler it is shown that if v r and v r are endowed with the necessary braid statistics then their braided tensor product v r v r is a realization of the braided matrices b r introduced previously while their inner product leads to an invariant quantum trace introducing braid statistics in this way leads to a fully covariant quantum braided linear algebra the braided groups obtained from b r act on themselves by conjugation in a way impossible

non coboundary poisson lie structures on the book group all possible poisson lie pl structures on the 3d real lie group generated by a dilation and two commuting translations are obtained their classification is fully performed by relating these pl groups to the corresponding lie bialgebra structures on the corresponding book lie algebra by construction all these poisson structures are quadratic poisson hopf algebras for which the group multiplication is a poisson map in contrast to the case of simple lie groups it turns out that most of the pl structures on the book group are non coboundary ones moreover from the viewpoint of poisson dynamics the most interesting pl book structures are just some of these non coboundaries which are explicitly analysed in particular we show that the two different q deformed poisson versions of the sl 2 r algebra appear as two distinguished cases in this classification as well as the quadratic poisson structure that underlies the integrability of a large class of 3d lotka volterra equations finally the quantization problem for these pl groups is sketched

quantum groups and deformation quantization explicit approaches and implicit aspects deformation quantization which gives a development of quantum mechanics independent of the operator algebra formulation and quantum groups which arose from the inverse scattering method and a study of yang baxter equations share a common idea abstracted earlier in algebraic deformation theory that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations i e quantizations in deformation quantization the algebraic object is the algebra of observables functions on symplectic phase space whose infinitesimal deformation is the poisson bracket and global deformation a star product in quantum groups it is a hopf algebra generally either of functions on a lie group or often its dual in the topological vector space sense as we briefly explain a completed universal enveloping algebra of a lie algebra with for infinitesimal a matrix satisfying the modified classical yang baxter equation mcybe frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract one success here comes from universal deformation formulas udfs expressions built from a lie algebra which deform any algebra on which the lie algebra operates as derivations the most famous of these is the moyal product a special case of a class in which the lie algebra is abelian another comes from recognition that the belavin drinfel d solutions to the mcybe are in fact infinitesimal deformations for which in the case of the special linear groups it is possible to give explicit formulas for the corresponding quantum yang baxter equations this review paper discusses necessarily in brief these and related topics including twisting as a form of udf and finding formulas for preferred deformations of hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation

quantifying the dynamical complexity of chaotic time series a powerful approach is proposed for the characterization of chaotic signals it is based on the combined use of two classes of indicators i the probability of suitable symbolic sequences obtained from the ordinal patterns of the corresponding time series ii the width of the corresponding cylinder sets this way much information can be extracted and used to quantify the complexity of a given signal as an example of the potentiality of the method i introduce a modified permutation entropy which allows for quantitative estimates of the kolmogorov sinai entropy in hyperchaotic models where other methods would be unpractical as a by product estimates of the fractal dimension of the underlying attractors are possible as well

on h k manifolds with asymptotic phase abstract we introduce the notion of h k manifolds and give conditions under which the property of being a manifold with asymptotic phase holds these conditions allow the construction of a transformation of the variational equation of a nonlinear nonautonomous systems with h k dichotomies into an almost decoupled quasilinear system the class of admissiblehandkfunctions is also briefly analyzed

quantization of poisson lie groups and applications letg be a connected poisson lie group we discuss aspects of the question of drinfel d can g be quantized and give some answers wheng is semisimple a case where the answer isyes we introduce quantizable poisson subalgebras ofc g related to harmonic analysis ong they are a generalization of f r t models of quantum groups and provide new examples of quantized poisson algebras

relativistic and newtonian κ space times the deformations of the galilei algebra and their associated noncommutative newtonian space times are investigated this is done by analyzing the possible nonrelativistic limits of an eleven generator pseudo extended poincare algebra p and their implications for the existence of a first order differential calculus the additional one form needed to achieve a consistent calculus on minkowski space is shown to be related to the additional central generator entering in the p hopf algebra in the process deformations of the extended galilei and galilei algebras are introduced which have respectively a cocycle and a bicrossproduct structure

classification of real three dimensional lie bialgebras and their poisson lie groups classical r matrices of the three dimensional real lie bialgebras are obtained in this way all three dimensional real coboundary lie bialgebras and their types triangular quasitriangular or factorizable are classified then by using the sklyanin bracket the poisson structures on the related poisson lie groups are obtained

twisting cocycle for null plane quantized poincare algebra the twisting cocycle for a quantized poincare algebra introduced by ballesteros et al is evaluated the solution is obtained as a specific case of a formulated multidimensional generalization of the non standard jordanian quantization of sl 2

extended jordanian twists for lie algebras jordanian quantizations of lie algebras are studied using the factorizable twists for a restricted borel subalgebra b of sl n the explicit expressions are obtained for the twist element f universal r matrix and the corresponding canonical element t it is shown that the twisted hopf algebra uf b is self dual the cohomological properties of the involved lie bialgebras are studied to justify the existence of a contraction from the dinfeld jimbo quantization to the jordanian one the construction of the twist is generalized to a certain type of inhomogenious lie algebras

deformation of orthosymplectic lie superalgebra osp 1 4 triangular deformation of the orthosymplectic lie superalgebra osp 14 is defined by chains of twists the corresponding classical r matrix is obtained by a contraction procedure from the trigonometric r matrix the carrier space of the constant r matrix and the twist element is the borel subalgebra the known super jordanian twist of osp 12 is generalized as an extended twist to the orthosymplectic lie superalgebras of higher rank the dimension of the cobracket kernel is related to the number of generators with primitive twisted coproduct of the deformed algebra

twisted classical poincare algebras we consider the twisting of the hopf structure for the classical enveloping algebra u g where g is an inhomogenous rotation algebra with explicit formulae given for the d 4 poincare algebra g p4 the comultiplications of twisted uf p4 are obtained by conjugating the primitive classical coproducts by f in u c x u c where c denotes any abelian subalgebra of p4 and the universal r matrices for uf p4 are triangular as an example we show that the quantum deformation of the poincare algebra recently proposed by chaichian and demiczev is a twisted classical poincare algebra the interpretation of the twisted poincare algebra as describing relativistic symmetries with clustered two particle states is proposed

quantumsu 2 ande 2 groups contraction procedure in 3 it was shown in the framework of deformed enveloping algebras that quantumsu 2 ande 2 groups are related by the contraction procedure we consider the same problem on thec level as a result we find a number of formulae coupling the comultiplications in quantumsu 2 ande 2 in particular we show that the comultiplications in both groups are implemented by partial isometries an unexpected feature of quantume 2 is discovered and the corresponding strange behavior of quantumsu 2 is described

integrable deformations of rossler and lorenz systems from poisson lie groups abstract a method to construct integrable deformations of hamiltonian systems of odes endowed with lie poisson symmetries is proposed by considering poisson lie groups as deformations of lie poisson co algebras moreover the underlying lie poisson symmetry of the initial system of odes is used to construct integrable coupled systems whose integrable deformations can be obtained through the construction of the appropriate poisson lie groups that deform the initial symmetry the approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of rossler and lorenz systems it is worth stressing that such deformations are of non polynomial type since they are obtained through an exponentiation process that gives rise to the poisson lie group from its infinitesimal lie bialgebra structure the full deformation procedure is essentially algorithmic and can be computerized to a large extent

classical deformations poisson lie contractions and quantization of dual lie bialgebras a poisson hopf algebra of smooth functions is simultaneously constructed on the two dimensional euclidean poincare and heisenberg groups by using a classical r matrix which is invariant under contraction the quantization for this algebra of functions is developed and its dual hopf algebra is also computed contractions on these quantum groups are studied it is shown that within this setting classical deformations are transformed into quantum ones by hopf algebra duality and the quantum heisenberg algebra is derived by means of a dual poisson lie quantization that deforms the standard moyal weyl product

visibility graphs and symbolic dynamics abstract visibility algorithms are a family of geometric and ordering criteria by which a real valued time series of n data is mapped into a graph of n nodes this graph has been shown to often inherit in its topology nontrivial properties of the series structure and can thus be seen as a combinatorial representation of a dynamical system here we explore in some detail the relation between visibility graphs and symbolic dynamics to do that we consider the degree sequence of horizontal visibility graphs generated by the one parameter logistic map for a range of values of the parameter for which the map shows chaotic behaviour numerically we observe that in the chaotic region the block entropies of these sequences systematically converge to the lyapunov exponent of the time series hence pesin s identity suggests that these block entropies are converging to the kolmogorov sinai entropy of the physical measure which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property to give analytical insight we explore the relation k x x 0 1 that for a given datum with value x assigns in graph space a node with degree k in the case of the out degree sequence such relation is indeed a piece wise constant function by making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs

jordanian twists on deformed carrier subspaces the nontrivial subspaces with primitive coproducts are found in the deformed universal enveloping algebras they can form carrier subspaces for additional jordanian twists the latter can be used to construct sequences of twists for algebras whose root systems contain long series of roots the corresponding twist for the so 5 algebra is given explicitly

invariant manifolds with asymptotic phase this paper is concerned with invariant manifolds allowing a generic approach to solutions of the underlying autonomous differential system only this means that any solution approaching such an invariant manifold m ultimately behaves like a particular solution on j4 in the sense that the difference n t i l of the two solutions tends to 0 as t cq in other words only solutions on the stable manifold of m can approach m as f cc invariant manifolds with that property are said to have an asymptotic phase the simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose floquet multipliers lie all but one in the open unit disc of the complex plane malkin 12 proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold more generally a k dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k floquet multipliers have modulus less than 1 see hale stokes 9 conditions for a torus to be asymptotically stable with asymptotic phase are given by samoilenko 13 and coppel 5 invariant manifolds without any specified structure are considered by fenichel 6 7 and in kirchgraber s note 111 moreover each center manifold has an asymptotic phase if the corresponding equilibrium is stable see carr al amood 4 and henry lo in any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold in each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability a negative answer to this question is given by the author in l 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase in l the manifold is supposed to be compact while the noncompact case is treated in 2 under the restriction that the generating family of periodic solutions has an amplitude independent period recently hale massatt 8 extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations in this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result a compact invariant manifold has an asymptotic phase if the manifold is hyperbolic assumption hl and carries a flow parallel in the sense of our assumption h2 this includes most of the aforementioned results

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

a nonlinear model of heat conduction a nonlinear model of heat propagation is presented from which a new heat conduction equation is derived an exact similarity solution in closed form of this equation is obtained which reveals the travelling wave characteristics for the transient temperature distribution it is shown that the temperature disturbances propagate with finite velocity which is a monotonically decreasing function of time

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

permutation entropy analysis based on gini simpson index for financial time series abstract in this paper a new coefficient is proposed with the objective of quantifying the level of complexity for financial time series for researching complexity measures from the view of entropy we propose a new permutation entropy based on gini simpson index gpe logistic map is applied to simulate time series to show the accuracy of the gpe method and expound the extreme robustness of our gpe by the results of simulated time series meanwhile we compare the effect of the different order of gpe and then we employ it to us and european and chinese stock markets in order to reveal the inner mechanism hidden in the original financial time series after comparison of these results of stock indexes it can be concluded that the relevance of different stock markets are obvious to study the complexity features and properties of financial time series it can provide valuable information for understanding the inner mechanism of financial markets

rational top and its classical r matrix we construct a rational integrable system the rational top on a co adjoint orbit of sl n lie group it is described by the lax operator with spectral parameter and classical non dynamical skew symmetric r matrix in the case of the orbit of minimal dimension the model is gauge equivalent to the rational calogero moser cm system to obtain the results we represent the lax operator of the cm model in two different factorized forms without spectral parameter related to the spinless case and another one with the spectral parameter the latter gives rise to the rational top while the first one is related to generalized cremmer gervais r matrices the gauge transformation relating the rational top and cm model provides the classical rational version of the irf vertex correspondence from the geometrical point of view it describes the modification of bundles over degenerated elliptic curve in view of the symplectic hecke correspondence the rational top is related to the rational spin cm model possible applications and generalizations of the suggested construction are discussed in particular the obtained r matrix defines a class of kzb equations

jordanian quantum algebra h sl n via contraction method and mapping using a contraction procedure developed earlier we construct in the first part of the present paper the jordanian quantum hopf algebra h sl 3 which has a remarkably simple coalgebraic structure and contains the jordanian algebra h sl 2 obtained by ohn as a hopf subalgebra a nonlinear map between the quantum h sl 3 and the classical sl 3 algebras is then established and the universal h matrix of the h sl 3 algebra is given in the second part we give the higher dimensional jordanian algebras h sl n for all n the universal h matrix of the h sl n algebra is also given

classical lie algebras and drinfeld doubles the drinfeld double structure underlying the cartan series an bn cn dn of simple lie algebras is discussed this structure is determined by two disjoint solvable subalgebras matched by a pairing for the two nilpotent positive and negative root subalgebras the pairing is natural and in the cartan subalgebra is defined with the help of a central extension of the algebra a new completely determined basis is found from the compatibility conditions in the double and a different perspective for quantization is presented other related drinfeld doubles on are also considered

poisson lie t duality for quasitriangular lie bialgebras we introduce a new 2 parameter family of sigma models exhibiting poisson lie t duality on a quasitriangular poisson lie group g the models contain previously known models as well as a new 1 parameter line of models having the novel feature that the lagrangian takes the simple form where the generalised metric e is constant not dependent on the field u as in previous models we characterise these models in terms of a global conserved g invariance the models on g su 2 and its dual g are computed explicitly the general theory of poisson lie t duality is also extended notably the reduction of the hamiltonian formulation to constant loops as integrable motion on the group manifold the approach also points in principle to the extension of t duality in the hamiltonian formulation to group factorisations d g m where the subgroups need not be dual or connected to the drinfeld double

classification of two and three dimensional lie superbialgebras using adjoint representation of lie superalgebras we obtain the matrix form of super jacobi and mixed super jacobi identities of lie superbialgebras by direct calculations of these identities and use of automorphism supergroups of two and three dimensional lie superalgebras we obtain and classify all two and three dimensional lie superbialgebras