the quantum heisenberg group h 1 q the structure of the quantum heisenberg group is studied in the two different frameworks of the lie algebra deformations and of the quantum matrix pseudogroups the r matrix connecting the two approaches together with its classical limit r are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed some unusual properties of the quantum enveloping heisenberg algebra are shown

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

dynamical systems and quantum bicrossproduct algebras we present a unified study of some aspects of quantum bicrossproduct algebras of inhomogeneous lie algebras such as poincare galilei and euclidean in n dimensions the action associated with the bicrossproduct structure allows us to obtain a nonlinear action over a new group linked to the translations this new nonlinear action associates a dynamical system with each generator which is the object of our study

classical r matrices via semidualisation we study the interplay between double cross sum decompositions of a given lie algebra and classical r matrices for its semidual for a class of lie algebras which can be obtained by a process of generalised complexification we derive an expression for classical r matrices of the semidual lie bialgebra in terms of the data which determines the decomposition of the original lie algebra applied to the local isometry lie algebras arising in three dimensional gravity decomposition and semidualisation yields the main class of non trivial r matrices for the euclidean and poincare group in three dimensions in addition the construction links the r matrices with the bianchi classification of three dimensional real lie algebras

poisson hopf limit of quantum algebras the poisson hopf analogue of an arbitrary quantum algebra uz g is constructed by introducing a one parameter family of quantizations uz g depending explicitly on and by taking the appropriate 0 limit the q poisson analogues of the su 2 algebra are discussed and the novel case is introduced the q serre relations are also extended to the poisson limit this approach opens the perspective for possible applications of higher rank q deformed hopf algebras in semiclassical contexts

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

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

inclusions irreductibles de facteurs et unitaires multiplicatifs ii resume from an irreducible depth 2 inclusion m 0 m 1 of factors with a faithful semifinite normal operator valued weight verifying a regularity condition we construct the canonical tower m i i n we prove then that both relative commutants m 2 m 0 and m 3 m 1 bear woronowicz algebra structures dual to each other as defined by masuda and nakagami moreover there exists an outer action of m 3 m 1 on m 1 such that m 0 is the fixed point algebra and m 2 is the crossed product the tower appears then as the tower of successive crossed products

state space modeling of two dimensional vector exponential trajectories we solve two problems in modeling polynomial vector exponential trajectories dependent on two independent variables in the first one we assume that the data generating system has no inputs and we compute a state representation of the most powerful unfalsified model for this data in the second instance we assume that the data generating system is controllable and quarter plane causal and we compute a roesser input state output model we provide procedures for solving these identification problems both based on the factorization of constant matrices directly constructed from the data from which state trajectories can be computed

a behavioral approach to control of distributed systems this paper develops a theory of control for distributed systems i e those defined by systems of constant coefficient partial differential operators via the behavioral approach of willems the study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates as in the lumped case behaviors of distributed arma systems can be reduced to ar behaviors this paper first studies the notion of ar controllable distributed systems following the corresponding definition for lumped systems due to willems it shows that as in the lumped case the class of controllable ar systems is precisely the class of ma systems it then shows that controllable 2 d distributed systems are necessarily given by free submodules whereas this is not the case for n d distributed systems n ge 3 this therefore points out an important difference between these two cases this paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems control is the process of restricting a behavior to a specific desirable autonomous subbehavior a notion of stability generalizing bounded input bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems this paper then solves this problem for a class of distributed behaviors

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

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

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

bicrossproducts of multiplier hopf algebras abstract in this paper we generalize majid s bicrossproduct construction we start with a pair a b of two regular multiplier hopf algebras we assume that b is a right a module algebra and that a is a left b comodule coalgebra the right action of a on b gives rise to the smash product a b the left coaction of b on a gives a possible coproduct on a b we discuss in detail the necessary compatibility conditions between the action and the coaction for to be a proper coproduct on a b the result is again a regular multiplier hopf algebra majid s construction is obtained when we have hopf algebras we also look at the dual case constructed from a pair c d of regular multiplier hopf algebras where now c is a left d module algebra while d is a right c comodule coalgebra we show that indeed these two constructions are dual to each other in the sense that a natural pairing of a with c and of b with d yields a duality between a b and the smash product c d we show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group i e a regular multiplier hopf algebra with integrals the algebra case is also considered some special cases are treated and they are related with other constructions available in the literature the basic example coming from a not necessarily finite group g with two subgroups h and k such that g k h and h k e where e is the identity of g is used to illustrate our theory more examples will be considered in forthcoming papers on the subject

quantum group gauge theory on quantum spaces we construct quantum group valued canonical connections on quantum homogeneous spaces including aq deformed dirac monopole on the quantum sphere of podles with quantum differential structure coming from the 3d calculus of woronowicz onsu q 2 the construction is presented within the setting of a general theory of quantum principal bundles with quantum group hopf algebra fibre associated quantum vector bundles and connection one forms both the base space spacetime and the total space are non commutative algebras quantum spaces

characteristic frequencies polynomial exponential trajectories and linear exact modeling with multidimensional behaviors the characteristic frequencies of a linear shift invariant multidimensional behavior correspond to its nonzero exponential trajectories the set of polynomial exponential trajectories belonging to a fixed characteristic frequency of a behavior is investigated a test is derived for determining whether this space is finite dimensional and if so a basis is constructed if it is infinite dimensional one considers only the polynomial exponential trajectories up to a certain degree of the polynomial part and a characterization is given of the asymptotic growth of the dimensions of these spaces as the degree bound tends to infinity a dual problem is concerned with linear exact modeling that is the construction of the so called most powerful unfalsified model mpum given a finite set of polynomial exponential trajectories the goal is to construct a behavior that contains the data and as little else as possible

time relevant stability of 2d systems for many 2d systems one of the independent variables plays a distinct role in the evolution of the trajectories since often this special independent variable is time we call such systems time relevant in this paper we introduce a stability notion for time relevant systems described by higher order difference equations we give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system we also give an lmi characterization of time relevant stability involving only constant matrices

lyapunov functions for time relevant 2d systems with application to first orthant stable systems we perform a lyapunov stability analysis of a special class of 2 d systems those for which one of the independent variables plays a distinguished role we show how to construct lyapunov functionals for this class using lmis we also apply our results to the study of quarter plane stability

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

green s functions on mumford curves we study an analogue of a green s function on mumford curves the meromorphic continuation of this green s function comes from a differential equation and we interpret the special value at s 0 as a volume of the corresponding curve using harmonic analysis on bruhat tits trees we derive a kronecker limit formula for the green s functions in question which connects the derivative at s 0 with the manin drinfeld theta functions following gross description of neron s local height pairing on mumford curves the special derivative here then equals twice of the neron s local height with sign changed

new time type and space type non standard quantum algebras and discrete symmetries starting from the classical r matrix of the non standard or jordanian quantum deformation of the sl 2 algebra new triangular quantum deformations for the real lie algebras so 2 2 so 3 1 and iso 2 1 are simultaneously constructed by using a graded contraction scheme these are realized as deformations of conformal algebras of 1 1 dimensional spacetimes time and space type quantum algebras are considered according to the generator that remains primitive after deformation either the time or the space translation respectively furthermore by introducing differential difference conformal realizations these families of quantum algebras are shown to be the symmetry algebras of either a time or a space discretization of 1 1 dimensional wave and laplace equations on uniform lattices the relationship with the known lie symmetry approach to these discrete equations is established by means of twist maps

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

quantum homogeneous spaces and special functions with a dimensional deformation parameter we study the most elementary aspects of harmonic analysis on a homogeneous space of a deformation of the two dimensional euclidean group admitting generalizations to dimensions three and four whose quantum parameter has the physical dimensions of length the homogeneous space is recognized as a new quantum plane and the action of the euclidean quantum group is used to determine an eigenvalue problem for the casimir operator which constitutes the analogue of the schrodinger equation in the presence of such a deformation the solutions are given in the plane wave and angular momentum bases and are expressed in terms of hypergeometric series with non commuting parameters

multidimensional behaviours an algebraic approach to control theory for pde this paper gives an overview of the algebraic theory of multidimensional behaviours and it contains some new aspects as well some known results are restated with a new proof after a short introduction to behavioural systems theory we briefly describe the mathematical background of the algebraic approach then we study structural properties such as autonomy and over determined systems on the one hand and controllability and flat systems on the other

quantum structure of the motion groups of the two dimensional cayley klein geometries a simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two dimensional cayley klein geometries their central extensions are also considered under this unified pattern in both cases some fundamental properties characterizing the classical ck geometries as the existence of a set of commuting involutions contractions and dualities relationships remain in the quantum version

graded contractions and kinematical groups of space time g graded contractions of the complex lie algebra b2 and of its real forms are considered it is found that a particular g z2 z2 is a proper choice of the grading group preserved when graded contractions leading to lie algebras of the kinematical groups of space time are studied

bicrossproduct structure of the null plane quantum poincare algebra a nonlinear change of basis allows us to show that the non standard quantum deformation of the 3 1 poincare algebra has a bicrossproduct structure quantum universal r matrix pauli lubanski and mass operators are presented in the new basis

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

twisting cocycles in fundamental representation and triangular bi crossproduct hopf algebras we find the general solution to the twisting equation in the tensor bialgebra t r of an associative unital ring r viewed as that of fundamental representation for a universal enveloping lie algebra and its quantum deformations we suggest a procedure of constructing twisting cocycles belonging to a given quasitriangular sub bialgebra h t r this algorithm generalizes reshetikhin s approach which involves cocycles fulfilling the yang baxter equation within this framework we study a class of quantized inhomogeneous lie algebras related to associative rings in a certain way for which we build twisting cocycles and universal r matrices our approach is a generalization of the methods developed for the case of commutative rings in our recent work including such well known examples as jordanian quantization of the borel subalgebra of sl 2 and the null plane quantized poincare algebra by ballesteros et al we reveal the role of special group 1 cocycles in this process and establish the bi crossproduct stru

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

mapping ideals of quantum group multipliers abstract we study the dual relationship between quantum group convolution maps l 1 g l g and completely bounded multipliers of g for a large class of locally compact quantum groups g we completely isomorphically identify the mapping ideal of row hilbert space factorizable convolution maps with m c b l 1 g yielding a quantum gilbert representation for completely bounded multipliers we also identify the mapping ideals of completely integral and completely nuclear convolution maps the latter coinciding with l 1 b g where b g is the quantum bohr compactification of g for quantum groups whose dual has bounded degree we show that the completely compact convolution maps coincide with c b g our techniques comprise a mixture of operator space theory and abstract harmonic analysis including fubini tensor products the non commutative grothendieck inequality quantum eberlein compactifications and a suitable notion of inner co amenable quantum group that we introduce and for which we exhibit examples from the bicrossed product construction our main results are new even in the setting of group von neumann algebras v n g for quasi sin locally compact groups g

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

cayley klein algebras as graded contractions of so n 1 we study z2 x n graded contractions of the real compact simple lie algebra so n 1 and we identify within them the cayley klein algebras as a naturally distinguished subset

contractions of the irreducible representations of the quantum algebras suq 2 and soq 3 the contractions of the irreducible representations of the unitary quantum algebra suq 2 and the orthogonal quantum algebra soq 3 in the gel fand tsetlin basis are regarded in detail with the help of the dual numbers

c statistical quantum groups and weyl algebras c statistical quantum groups and algebras are obtained by a process of transmutation they are like super quantum groups and algebras but with statistics given now by a complex number among the examples it is shown that the weyl algebra of canonical quantization may be viewed equivalently as a c statistical plane related examples are the noncommutative torus and a double loop variable quantization of photons these results lead toward a reformulation of ordinary quantum mechanics as a classical theory with c statistics in this reformulation the role of the 1 factors of super statistics is played by the free particle time evolution

on the takagi interpolation problem in this paper we approach the interpolation problem of takagi from the perspective of time series modeling the framework of exact identification and the notion of mpum are used in order to generalize some results on the nevanlinna interpolation problem these results are of independent interest and concern the relation between the roots of the determinant of the denominator of an unfalsified model and the signature of the pick matrix of the data

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

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

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

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given