poincare bendixson theory for leaves of codimension one the level of a local minimal set of a c2 codimension one foliation of a compact manifold is a nonnegative integer defined inductively level zero corresponding to the minimal sets in the usual sense each leaf of a local minimal set at level k is at level k the authors develop a theory of local minimal sets level and how leaves at level k asymptotically approach leaves at lower level this last generalizes the classical poincare bendixson theorem and provides information relating growth topological type and level e g if l is a totally proper leaf at level k then l has exactly polynomial growth of degree k and topological type k

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

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

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

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

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

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

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

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

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

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

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

distributed algorithms for reaching consensus on general functions this paper presents analysis and design results for distributed consensus algorithms in multi agent networks we consider continuous consensus functions of the initial state of the network agents under mild smoothness assumptions we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus this characterization is the building block to obtain various design results for networks with weighted directed interconnection topologies we first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus we apply this result to the family of weighted power mean functions and characterize the exponential convergence properties of the resulting algorithms we establish the validity of these results for scenarios with switching interconnection topologies finally we conclude with two discontinuous distributed algorithms that achieve respectively max and min consensus in finite time

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

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

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

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